Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Sample 1:
68,64,72,70,68,68,69,72,68,78,66,67,72,71,72,69,70,64,74,72,73,70,69,78,70,68,69,
71,66,75,70,68,68,67,70,73,68,70,71,67,68,70,73,68,67,73,69,67,64,77
Sample 2:
71,71,69,71,71,70,74,71,75,77,76,69,65,73,67,71,71,62,75,75,69,71,67,68,71,71,66,
72,64,70,68,72,72,68,71,70,66,67,72,69,68,68,72,64,70,67,73,73,70,68
Sample 3:
68,69,72,68,67,71,74,72,72,65,64,67,74,65,66,67,72,67,72,66,71,71,73,68,72,72,67,
68,71,68,70,66,73,72,73,70,68,67,67,70,72,71,68,69,69,65,68,70,72,75
Sample 4:
69,75,68,70,72,71,70,74,67,68,68,69,64,69,68,70,70,75,65,69,72,64,73,75,70,67,71,
70,71,69,67,70,70,68,67,67,72,71,65,62,66,64,70,69,70,73,71,68,72,66
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
75,65,71,78,70,70,75,71,71,73,72,73,70,70,73,68,71,70,70,67,70,70,72,67,66,70,70,
69,64,70,65,70,69,72,68,70,62,73,68,63,71,70,71,67,71,71,71,68,69,68
67,73,73,69,66,69,71,71,67,63,70,68,68,70,70,68,71,69,65,76,69,68,71,64,67,69,68,
65,65,70,71,75,72,71,67,71,64,70,73,66,73,73,70,72,68,70,65,75,75,66
68,70,63,72,72,68,71,69,70,73,65,69,66,72,70,72,78,69,73,67,68,68,72,70,70,68,70,
71,71,71,70,71,64,70,72,67,70,74,71,70,67,69,76,66,63,70,68,71,70,71
69,70,64,68,71,71,77,66,74,68,72,69,71,72,73,72,72,73,75,74,66,69,72,71,63,71,71,
71,73,73,72,70,70,73,75,66,71,67,71,70,71,71,64,69,73,72,73,71,68,71
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
69,72,64,68,74,68,74,68,71,74,70,72,71,72,70,73,67,71,63,68,69,60,71,68,66,72,67,
67,74,68,67,65,70,76,73,63,66,71,68,67,64,70,72,71,68,68,66,71,70,67
74,73,72,68,67,71,72,71,67,67,67,73,74,70,66,66,66,71,72,72,70,67,72,68,64,65,68,
73,67,71,67,65,71,74,74,72,71,68,67,74,72,71,73,70,70,74,69,71,68,65
76,66,71,71,67,73,71,66,71,67,71,69,70,75,72,69,64,63,68,65,69,72,67,65,68,73,67,
64,71,68,68,71,67,64,70,66,74,67,76,73,66,68,68,62,73,71,66,71,72,61
66,74,67,67,71,62,68,71,67,77,72,68,71,68,71,72,75,69,68,74,70,70,70,75,71,70,74,
68,71,64,66,67,65,71,67,70,72,61,69,69,72,75,71,67,66,67,66,70,74,66
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
67,66,74,64,72,75,69,71,76,71,76,72,70,70,69,65,67,70,67,62,70,68,71,70,77,69,65,
68,69,68,78,69,61,66,72,70,64,72,69,80,71,71,69,68,74,68,67,75,73,65
71,66,70,70,72,67,75,70,64,69,69,72,74,68,68,76,73,67,74,71,72,68,67,68,69,64,75,
67,66,74,71,67,71,63,75,70,67,75,69,68,68,70,66,68,72,74,72,69,70,72
68,67,64,68,70,74,63,70,67,66,69,71,73,69,68,70,71,69,71,71,67,66,70,71,66,69,68,
67,66,68,67,70,73,66,71,67,67,71,70,73,65,66,70,65,70,67,70,67,65,73
76,72,66,70,69,74,67,70,72,73,68,80,70,72,69,68,65,68,69,67,69,67,69,72,68,69,69,
71,72,73,71,70,68,63,72,69,68,74,68,69,74,69,75,69,68,66,71,68,67,63
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
75,66,65,70,72,65,69,69,67,71,63,70,70,66,67,73,76,72,71,66,70,68,73,66,71,70,73,
67,64,71,73,75,77,66,65,69,67,70,66,64,70,65,68,69,73,73,68,73,71,64
71,71,73,72,76,70,70,69,70,62,71,71,75,70,68,71,71,68,73,71,68,69,71,70,73,68,72,
70,69,68,72,75,67,74,71,74,67,66,71,76,75,69,72,74,71,71,69,72,69,68
71,71,68,72,72,81,64,72,72,67,63,69,69,67,65,70,75,74,67,66,70,71,73,68,68,72,68,
68,70,76,72,73,65,68,69,66,76,67,74,66,70,66,66,70,71,77,72,73,74,73
68,70,66,65,70,69,71,69,74,79,69,71,72,72,68,67,71,69,67,69,64,69,71,75,69,75,72,
69,73,72,71,73,69,78,71,69,68,67,70,70,64,69,67,72,78,69,65,69,72,74
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
72,71,68,65,68,71,72,61,67,69,69,68,67,63,67,64,70,72,72,60,71,69,73,70,73,73,66,
76,68,72,65,66,70,71,74,73,69,73,67,76,75,70,66,69,71,72,67,70,65,74
73,66,67,71,71,71,66,68,69,63,72,73,67,63,71,66,66,77,75,63,71,67,69,66,70,69,74,
68,65,67,67,74,72,73,67,71,70,68,70,72,69,68,65,64,73,69,73,64,72,68
70,73,72,70,68,74,69,70,68,67,61,75,72,75,74,67,70,70,74,69,69,73,70,69,70,71,73,
72,71,67,73,66,70,69,74,66,64,70,73,71,72,75,66,65,69,71,75,73,72,67
76,70,71,66,70,73,66,73,69,68,72,70,72,69,73,71,70,74,69,73,68,70,67,71,68,71,70,
70,67,72,69,64,67,69,66,72,66,65,75,69,67,74,67,67,75,64,67,69,75,66
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
72,72,70,73,72,66,75,74,71,72,68,69,73,63,73,71,65,66,65,64,68,71,71,76,68,68,71,
70,75,73,70,80,65,67,70,61,71,69,65,70,71,74,68,68,69,70,71,70,70,71
69,66,71,70,72,68,71,66,68,69,65,68,67,64,72,67,69,75,62,72,64,70,79,70,73,69,73,
63,67,69,71,71,68,70,70,68,71,64,70,76,72,71,64,71,66,72,66,73,70,72
71,70,70,70,71,71,69,72,72,70,69,68,73,69,71,72,70,66,67,71,73,72,70,71,68,68,68,
73,70,75,66,67,65,65,73,75,66,67,68,68,73,72,68,67,69,69,72,69,64,65
73,72,66,70,74,69,67,70,71,69,70,64,70,70,72,72,67,69,73,68,72,74,69,73,69,70,71,
70,76,69,70,70,72,67,68,71,73,72,71,69,71,67,67,67,64,68,70,71,68,69
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
68,73,71,72,75,78,70,69,68,66,68,69,69,71,67,69,71,68,68,69,66,68,72,71,66,75,72,
68,73,72,68,70,69,66,67,72,69,75,70,68,69,70,67,70,74,70,66,67,65,68
69,69,67,74,63,71,75,70,70,73,65,67,75,73,72,66,66,66,70,77,65,71,64,67,71,69,71,
70,69,70,65,63,64,70,74,69,70,71,68,68,70,66,68,68,66,63,71,68,69,71
77,64,66,69,64,71,70,69,73,68,73,74,74,71,66,72,68,65,70,65,68,72,73,69,70,67,65,
69,71,73,71,67,71,74,71,68,70,70,64,69,70,70,73,66,70,73,74,71,67,73
68,66,66,64,67,67,71,70,73,68,70,70,62,69,71,69,70,72,62,71,72,69,67,67,71,74,68,
75,77,68,67,71,67,76,63,72,68,73,66,71,67,73,69,68,72,71,69,69,65,73
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
73,70,72,73,63,65,66,70,72,70,67,70,71,73,67,66,68,65,74,70,72,76,71,72,69,72,63,
66,74,69,68,70,74,64,71,77,64,72,68,70,71,66,71,67,71,71,67,64,69,71
72,72,73,67,68,72,70,67,69,69,73,74,67,76,66,70,71,65,76,71,71,66,70,74,67,68,68,
68,65,65,65,64,70,71,64,69,69,62,69,67,66,69,75,72,71,72,66,68,67,68
75,71,64,73,68,69,72,66,68,68,71,64,69,70,68,71,68,73,69,69,72,68,69,69,69,67,67,
70,70,70,68,68,70,67,72,74,64,64,73,75,63,65,72,70,67,70,66,79,74,68
71,68,69,68,70,72,65,73,70,70,65,71,67,69,65,65,67,70,73,70,75,70,67,73,65,74,69,
75,73,70,71,69,67,66,74,68,69,67,68,67,70,71,70,71,69,74,71,71,77,66
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
62,77,70,74,70,67,73,69,68,76,71,66,75,73,70,75,72,71,67,69,64,70,71,67,68,70,71,
70,65,71,75,70,71,70,69,68,70,67,71,68,70,73,73,68,72,66,71,70,70,73
67,70,68,67,67,72,77,74,68,71,64,67,73,68,73,63,71,67,71,64,71,69,73,70,77,70,72,
73,70,71,66,64,70,68,67,72,70,64,72,70,70,73,77,76,67,69,67,68,70,67
71,72,72,66,67,64,65,67,72,67,66,70,71,65,69,64,71,69,76,68,69,68,72,69,75,71,69,
72,69,67,71,70,70,70,73,67,65,67,70,64,75,66,68,70,66,73,72,64,63,63
70,68,70,73,72,75,63,68,74,67,70,72,72,69,71,73,72,70,74,70,67,72,74,75,72,67,70,
72,70,72,71,74,68,72,68,73,71,67,70,66,73,67,72,67,67,69,66,70,74,67
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
76,66,69,65,68,72,69,74,71,69,71,67,71,70,70,71,73,70,67,70,72,69,65,70,71,73,73,
74,67,78,73,64,72,67,67,64,66,73,68,65,75,69,67,65,72,72,74,60,68,74
74,72,68,72,65,70,72,74,76,64,70,71,72,70,73,68,73,75,66,71,69,67,70,72,72,77,74,
77,64,67,74,71,62,69,65,62,70,68,72,61,70,68,66,70,69,64,63,67,71,70
71,73,72,71,70,66,69,66,65,75,63,74,70,66,69,70,70,68,70,77,73,75,65,68,71,62,70,
70,66,72,66,74,70,64,65,69,71,65,75,76,72,68,67,68,70,71,69,65,69,68
70,75,72,69,70,68,69,70,70,70,72,67,69,68,70,72,68,69,69,64,66,72,70,67,72,70,72,
69,75,68,70,71,73,68,70,71,75,68,71,67,68,61,69,65,64,73,72,69,71,70
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
65,68,71,68,68,70,70,74,70,66,74,66,71,67,77,67,69,72,69,72,67,74,71,71,73,73,67,
70,69,67,73,71,69,73,68,74,63,67,70,72,69,72,63,68,67,62,70,73,69,71
75,64,69,76,72,73,66,74,75,70,73,69,69,73,73,72,73,73,74,70,71,69,67,67,66,67,75,
69,66,66,71,68,71,64,72,70,72,70,70,70,64,69,64,64,67,69,70,71,70,71
70,75,72,68,70,66,64,65,71,65,68,72,68,69,67,72,66,63,70,69,67,76,74,65,69,70,67,
62,74,71,74,69,68,71,69,73,69,69,73,66,70,72,69,72,65,65,68,68,66,69
65,70,67,69,69,74,75,75,68,71,68,70,69,72,73,69,68,68,75,67,70,72,68,69,69,70,67,
69,72,70,73,66,70,65,69,70,69,65,68,69,72,70,71,76,73,66,76,64,70,77
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
71,66,70,66,69,78,70,72,71,69,66,67,69,71,67,70,73,67,69,69,68,64,71,75,68,68,71,
70,67,72,77,69,68,70,64,72,68,71,68,71,66,60,72,72,73,71,68,68,65,73
70,68,74,67,71,68,66,70,72,70,70,70,70,69,70,67,72,69,69,72,73,65,68,68,66,72,69,
68,73,69,61,68,73,68,67,66,70,68,71,66,69,69,71,68,64,73,72,66,69,62
68,67,69,68,68,71,73,69,70,71,70,70,78,63,70,67,68,71,72,71,71,71,67,70,74,73,64,
69,73,66,68,67,71,69,70,69,69,68,69,73,67,72,69,74,70,69,73,71,70,66
71,73,67,75,66,67,65,66,63,72,67,68,66,71,72,68,69,71,71,64,68,69,67,68,72,70,73,
73,67,69,68,74,74,64,68,68,68,67,72,73,67,73,71,70,66,68,68,68,69,69
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
67,71,76,68,69,64,69,70,73,67,72,66,66,72,68,73,69,73,69,72,67,69,65,63,67,71,72,
70,71,69,65,69,69,67,74,69,68,72,65,69,72,68,77,62,69,75,74,74,71,71
65,70,69,67,70,74,72,69,58,65,72,71,70,67,71,69,69,71,64,70,68,73,74,70,66,68,75,
69,68,71,68,72,69,69,64,77,71,69,74,71,70,71,76,69,70,69,71,69,72,69
64,70,71,68,71,72,67,77,70,69,72,70,69,69,75,68,62,71,66,76,77,75,62,75,71,73,70,
67,74,66,74,71,71,70,69,65,71,65,67,66,72,70,69,70,75,71,65,65,64,69
68,67,67,72,69,68,68,70,65,67,67,68,67,68,63,66,68,73,71,66,69,71,65,68,67,69,72,
68,65,64,70,66,74,69,70,70,73,76,70,71,71,72,74,71,71,68,68,70,65,68
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
67,73,70,65,68,70,70,69,68,71,65,67,71,75,66,66,70,71,71,67,69,70,63,66,66,71,66,
69,68,71,73,63,62,67,67,68,71,68,68,69,72,76,68,69,67,71,69,72,68,65
71,71,71,72,71,69,76,69,74,64,72,68,72,69,73,70,77,65,65,64,69,77,65,66,69,73,75,
70,72,67,75,65,64,74,70,71,69,70,72,75,62,67,71,73,72,66,72,68,71,76
65,69,70,65,69,65,71,68,68,72,74,67,66,70,69,73,65,70,70,70,74,69,61,71,67,69,77,
71,69,68,69,70,74,69,75,75,66,71,65,66,66,76,70,65,72,70,72,71,71,69
69,69,71,69,72,60,68,76,76,70,65,66,68,66,69,73,67,64,71,73,64,68,71,68,72,69,68,
75,71,71,68,69,76,67,66,72,71,65,66,76,69,70,72,77,70,67,72,71,68,77
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
65,69,73,64,69,69,69,72,72,66,62,74,65,67,72,71,68,73,66,68,73,68,70,73,77,70,67,
74,75,70,65,72,69,67,68,67,69,73,75,70,71,67,66,74,77,69,65,71,71,67
75,67,65,71,69,68,69,70,69,68,74,69,70,65,77,69,66,63,66,68,67,67,66,69,64,69,73,
66,68,70,69,68,74,69,65,65,70,70,66,71,73,67,71,69,69,72,65,68,76,69
69,72,70,72,69,68,74,73,67,72,69,64,70,72,65,68,67,68,73,69,69,70,65,63,72,65,66,
75,66,73,67,65,76,72,72,72,72,70,67,74,67,70,70,70,68,76,74,69,66,65
73,74,63,67,67,67,69,69,68,77,73,68,72,71,67,71,63,70,64,69,72,67,65,68,71,71,70,
66,74,73,69,70,70,68,66,66,68,74,70,73,68,70,71,67,71,68,72,69,67,67
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
71,77,70,71,67,64,67,69,71,68,73,73,66,67,69,72,69,69,71,73,67,64,69,72,67,65,69,
78,68,70,69,73,68,72,67,67,68,70,67,67,69,71,66,67,65,71,71,69,64,72
72,71,76,70,66,68,70,68,62,74,66,68,70,79,70,73,68,76,73,70,71,65,63,73,73,69,72,
70,71,68,67,70,68,70,74,71,69,64,70,67,72,72,68,70,68,68,74,65,70,72
66,67,73,73,75,70,69,68,71,67,68,75,69,72,70,63,71,70,64,68,69,71,71,74,67,65,63,
68,68,66,66,69,69,68,72,65,70,69,73,67,70,71,65,73,68,70,75,69,72,67
74,70,69,70,69,66,72,75,70,70,67,67,74,71,69,69,70,68,69,72,65,73,69,73,63,74,70,
63,64,71,68,68,72,67,67,74,76,73,70,73,73,66,67,72,74,78,67,66,68,70
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
74,66,68,73,73,71,73,72,72,69,70,67,70,76,73,66,73,71,66,65,70,73,68,73,66,70,70,
68,66,77,70,74,69,70,73,70,67,73,75,69,67,67,66,76,72,70,70,70,73,68
63,68,71,69,71,69,73,70,76,71,73,66,68,62,67,68,75,66,70,71,71,77,72,76,71,71,64,
74,74,73,67,72,76,69,67,69,74,70,65,68,71,67,68,68,71,68,71,66,69,72
69,67,73,67,75,69,72,75,68,68,70,73,69,69,70,75,70,71,68,69,72,66,71,74,71,69,69,
68,68,70,71,68,68,67,70,72,69,64,69,70,71,68,70,69,72,73,70,74,70,67
67,67,70,69,72,65,72,70,66,65,63,68,68,69,73,67,66,70,64,76,68,74,68,70,73,75,72,
68,69,66,69,65,67,67,75,72,72,73,67,73,64,71,72,61,72,72,70,72,73,74
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
70,71,75,69,66,67,76,70,73,69,73,71,74,70,71,64,70,68,71,67,69,70,70,68,71,68,65,
65,70,66,69,67,71,71,77,69,70,69,71,69,70,71,66,69,70,65,70,68,71,65
67,66,67,72,68,71,66,65,72,71,70,64,68,66,68,70,74,68,71,71,71,68,69,69,74,73,69,
66,73,68,72,71,70,71,68,73,68,70,65,70,75,69,70,70,65,67,73,76,71,70
70,72,63,69,72,67,68,71,69,68,68,68,64,71,70,69,69,66,68,70,71,68,67,72,63,73,70,
68,70,58,77,66,71,69,69,71,69,74,64,62,66,66,73,70,64,68,61,73,74,65
71,72,67,67,70,70,64,65,68,69,63,72,69,62,68,68,71,63,71,66,68,72,73,73,69,68,68,
71,69,71,72,68,63,72,73,73,73,71,68,67,67,68,73,67,70,69,69,72,67,67
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
70,73,69,69,77,76,72,66,70,71,71,65,73,70,68,72,66,68,65,72,65,72,73,69,68,71,75,
63,67,73,72,69,69,68,71,70,74,58,64,66,66,71,64,65,75,70,69,63,68,66
69,69,68,73,69,72,70,73,66,71,70,70,72,71,74,71,74,72,66,62,70,64,70,74,69,70,72,
72,71,70,70,67,64,69,69,69,73,64,67,67,73,69,72,68,72,68,68,65,70,74
65,71,67,68,65,73,69,71,70,67,71,71,70,69,69,71,71,67,66,63,71,73,66,67,73,76,68,
68,63,68,70,68,70,73,69,75,68,67,68,67,68,74,69,77,73,71,70,67,68,69
70,75,74,67,71,77,64,67,67,69,66,75,75,71,68,69,68,75,69,65,74,72,66,68,67,66,71,
72,66,74,71,69,72,68,67,67,65,65,64,68,72,67,74,67,75,71,71,68,68,71
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
68,70,64,73,67,73,70,68,67,72,67,72,68,67,67,71,64,67,68,74,65,69,70,68,70,64,73,
69,72,68,68,68,73,63,62,72,71,70,71,68,64,70,72,70,68,67,69,65,72,65
65,69,68,70,70,71,69,70,73,71,67,73,70,71,70,69,67,76,76,73,68,66,69,67,63,70,70,
68,71,68,65,65,72,65,69,76,71,70,70,72,64,73,68,66,67,71,70,72,68,66
69,75,66,68,73,76,65,70,70,65,75,68,67,68,65,70,66,67,69,65,71,67,67,72,67,66,71,
71,65,72,71,72,71,70,66,68,61,67,67,74,73,67,70,73,73,66,64,68,71,71
69,61,68,67,70,67,66,64,74,74,67,68,67,64,64,71,68,70,71,69,69,66,70,69,73,68,72,
71,71,70,67,67,72,69,69,72,70,73,65,65,71,69,71,72,67,72,66,69,69,69
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
70,68,67,65,70,67,67,66,66,68,69,70,68,69,70,71,71,74,71,67,76,69,66,75,64,71,72,
76,71,67,71,73,69,78,77,71,76,67,69,75,70,66,62,70,77,69,68,68,68,72
73,77,71,70,69,76,69,64,70,67,67,67,72,68,65,67,72,73,71,70,68,68,71,69,74,65,65,
65,66,72,67,68,69,65,70,68,69,68,69,74,73,67,68,69,72,72,69,69,71,73
71,68,66,68,69,71,64,69,66,65,72,73,76,66,68,68,70,67,70,76,62,69,64,75,68,73,67,
67,69,68,65,69,73,69,74,69,67,67,63,70,70,65,68,74,67,66,66,74,70,67
74,68,67,71,69,69,69,64,71,69,69,72,72,71,72,66,70,72,73,67,72,70,62,62,71,71,68,
66,75,74,65,74,73,70,66,71,66,67,74,75,70,67,66,71,68,72,66,64,63,66
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
67,66,70,65,61,71,71,72,74,69,70,73,66,70,72,73,71,66,66,76,65,69,69,71,66,74,64,
68,68,67,70,70,68,65,74,77,64,70,71,70,67,71,62,69,71,69,71,66,68,71
67,72,73,74,71,69,68,66,73,68,73,74,69,69,74,69,78,64,75,68,74,64,66,70,72,69,65,
68,65,68,76,68,64,73,64,72,68,65,64,66,70,69,72,70,72,69,70,68,66,69
69,63,64,71,73,69,67,66,63,67,72,69,68,68,66,67,71,72,72,67,71,68,74,76,69,72,71,
68,70,67,70,67,62,72,69,69,70,69,76,73,66,71,68,74,73,69,74,71,70,70
74,67,75,64,65,68,70,75,72,64,72,68,69,75,72,72,70,64,66,71,68,68,74,70,74,66,71,
68,68,71,71,68,64,74,70,63,66,72,69,77,67,73,73,71,71,72,71,66,67,67
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
74,71,67,73,65,68,73,76,70,79,63,71,73,73,71,68,75,63,73,72,66,64,68,75,68,70,72,
68,68,74,70,71,66,72,71,70,70,72,65,67,68,71,66,70,66,75,66,68,69,66
69,69,72,68,71,66,68,67,71,74,70,73,71,69,74,71,70,69,73,70,75,72,66,76,67,73,71,
69,71,71,70,72,71,58,69,70,72,63,67,66,70,72,73,71,70,72,62,72,74,71
68,71,73,70,65,69,68,69,68,68,73,72,73,70,70,74,67,74,75,65,70,63,69,68,67,67,67,
71,69,72,77,70,67,66,66,67,68,71,70,73,75,72,72,69,65,70,70,66,75,74
65,70,74,70,64,68,65,76,64,69,71,73,71,74,72,73,72,73,68,67,68,71,68,66,76,70,69,
73,68,74,67,68,69,70,66,73,71,68,65,69,71,77,71,66,71,67,69,73,72,68
Math 102 - 04
Exercise on Sampling Confidence Intervals
This is an in-class exercise on sampling confidence intervals.
Our population consists of all adult males in the US. We are measuring their height
in inches and want to determine the proportion of the US adult male population that
is at least 6 feet tall. We let p be our population proportion which is currently
unknown.
We will estimate p with a sample, 𝑝̂ . We take a sample that has n=50 US adult males
and measure their height. We will use this sample to build 90% confidence intervals
for the unknown p.
You have four samples provided below. For each sample, follow the instructions.
1. First circle those individuals in your sample with a height of at least 6 feet
(72 inches or greater). Count the number that meet this criteria and divide
by n=50 to determine your 𝑝̂ . Do this for each sample.
2. Use your value for 𝑝̂ to estimate the population variance 𝜎𝑝̂ .
3. Use your estimate of 𝜎𝑝̂ to compute the upper and lower bounds on p using a
90% confidence interval.
4. If your professor tells you that the true value of p = 0.2638, then determine
which of your confidence intervals capture the true value of p.
Samples:
72,72,68,67,72,74,66,68,72,71,72,74,67,75,73,69,68,65,75,67,67,69,74,72,68,75,67,
68,66,69,69,71,71,67,67,71,71,71,68,65,71,72,66,71,71,70,71,71,64,73
70,65,70,60,67,70,71,75,68,69,76,70,72,73,68,69,70,70,70,68,69,68,68,67,64,67,72,
75,67,73,67,72,65,70,66,71,69,70,69,73,71,65,69,64,70,66,72,71,74,74
66,70,69,71,73,73,69,72,68,68,73,73,65,68,68,76,72,72,65,67,74,72,73,69,73,66,65,
72,66,69,72,68,67,69,68,68,76,68,68,69,73,70,68,67,67,69,69,69,73,73
68,64,68,68,65,69,72,70,70,72,66,69,72,68,65,69,72,73,67,71,76,69,73,68,70,72,72,
73,68,76,70,74,67,68,66,67,65,69,70,67,71,72,66,73,74,73,70,66,70,74