Answers for Practice Questions for Exam III
1.
If X is a Binomial random variable and if np ≥ 10 and n(1-p) ≥ 10, then X is approximately NORMAL with µX = np
σX =
2.
X is a Binomial random variable with n = 50 and p = 0.15. Should the Normal approximation be used to find P(X
> 10)? Why or why not? No. np < 10.
3.
Let X have a Binomial distribution with n = 379 and p = 0.09. If appropriate, use the Normal approximation to
find P(X > 40). If not, explain why not.
4.
np = 34.11 > 10 P(X > 40) ≈ 14.46%
The Bureau of Labor Statistics reported in 2010 that 17% of U.S. nonfarm workers are government employees.
A random sample of 50 workers is drawn.
a.
Is it appropriate to use the Normal approximation to find the probability that less than 20% of workers
in the sample are government employees? If so, find the probability. If not, explain why not.
When n = 50, the Normal approximation cannot be used because np > 10, so the distribution of phat is NOT approximately Normal.
b.
Estimate the probability that less than 20% of workers in a sample of size 150 are government
employees.
When n = 150, the Normal approximation CAN be used because np > 10, so the distribution of p-hat
IS approximately Normal.
5.
( < 0.2)
A SRS of size n = 111 is taken from a population with proportion of success p = 0.54.
Estimate the probability that the sample proportion will be between 46% and 59%. That is, find
0.59.
6.
81%
(the two z-scores are z = - 1.69 and z = 1.06)
The CENTRAL LIMIT THEOREM states that for non-Normal populations, the sampling distribution of
is
approximately Normal when the sample size is large enough (and “large enough” means AT LEAST 30.
7.
A sample of size 75 will be drawn from a population with mean µ = 10 and standard deviation
σ = 12.
8.
a.
Find the probability that
will be between 8 and 14. P(8 <
b.
Find the 15 percentile of . For the 15 percentile, z = - 1.04.
th
< 14) ≈ 92.32%
th
= 8.6
An IQ test was given to a SRS of 65 students at a certain college. The sample mean score was 105.2. Scores on
this test are known to have a standard deviation of σ = 10. We want to construct a 90% CI estimate for the
mean IQ score of students at this college.
a.
Find the margin of error. m = 2.04 or 2.0.
b.
Give the interval and write a sentence or two that interprets it.
We estimate with 90% confidence that the mean IQ score of students at this college is between
103.2 and 107.2. (The method used to get the interval is accurate for about 90% of all randomly
chosen samples.)
c.
Is it likely that the population mean is greater than 100? Explain your reasoning.
Practice Questions for Exam III
Fall 2011
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Yes, I’d say so – our interval is well above 100.
d.
Using the same data, construct a 95% CI for the mean IQ score. (102.8, 107.6)
e.
If you were to construct a 98% CI for the mean IQ score (using the same data) would your margin of
error be less than it was for the 95% CI or would it be greater? Explain. It would be greater. When
the confidence level goes up, the margin of error increases and the interval is wider.
f.
How many students must be in the sample if you want 98% confidence and a margin of error no more
than 3 points?
9.
For 98% confidence, z* = 2.326. We need a sample size of n = 61.
A survey organization drew a SRS of 625 households from a city of 100,000 households. The mean number of
people in the 625 households was 2.30 and a 95% CI for the mean number of people in the 100,000 households
was (2.16, 2.44). Which of the following statements is (are) true:
a.
We are 95% confident that the mean number of people in the 625 households is between 2.16 and
2.44.
b.
FALSE
We are 95% confident that the mean number of people in the 100,000 households is between 2.16
and 2.44.
c.
TRUE
The probability is 0.95 that the mean number of people in the 100,000 households is between 2.16
and 2.44.
FALSE
d.
95% of the households in the sample contain between 2.16 and 2.44 people.
e.
95% of the households in the population (the 100,000 households) contain between 2.16 and 2.44
people
FALSE
FALSE
10. In the CI 24.3 ± 2.1, the quantity 2.1 is called the MARGIN OF ERROR.
11. If we increase the confidence level and keep the sample size the same, what happens to the margin of error?
The margin of error increases. (To be more confident that the interval “captures” the parameter value, we
need a wider interval.)
12. TRUE or FALSE: The confidence level is the proportion of all possible samples for which the confidence interval
will “capture” or “cover” the true parameter value. TRUE
13. A same of size 10 is drawn from a Normal population whose standard deviation is σ = 2.5. The sample mean is
7.92.
a.
Construct a 95% CI for the population mean µ.
(6.37, 9.47)
b.
If the population were not Normal, would the CI constructed in part (a) be valid? Explain.
No. For non-Normal populations, CIs are appropriate only if n > 30, and then they are approximate.
14. According to the National Health Statistics Reports, a sample of 360 one-year old baby boys in the U.S. had a
mean weight of 25.5 pounds. Assume the population standard deviation is 5.3 pounds.
a.
Construct a 95% CI for the mean weight of all one-year-old baby boys in the U.S.
Based on this sample, we can estimate that the mean weight of all one-year-old baby boys in the
U.S. is 25.5 pounds, with a 95% margin of error of 0.5 pounds.
b.
Should this CI be used to estimate the mean weight of all one-year-old babies in the U.S.? Why or why
not?
No, because our data was only for baby BOYS.
Practice Questions for Exam III
Fall 2011
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15. The NULL hypothesis states that a parameter is equal to a certain value.
16. TRUE or FALSE: If we reject Ho, we conclude that Ho is false.
17. TRUE or FALSE: If we do not reject Ho, then we conclude that Ha is false.
18. TRUE or FALSE: If we do not reject Ho, then we conclude that Ho is true.
19. Determine whether the outcome is a Type I error, a Type II error, or a correct decision:
a.
A test is made of Ho: µ = 20 versus Ha: µ ≠ 20. The true value of µ is 25, and Ho is rejected.
Correct decision.
b.
A test is made of Ho: µ = 5 versus Ha: µ < 5. The true value of µ is 5, and Ho is rejected.
Type I error.
c.
A test is made of Ho: µ = 63 versus Ha: µ > 63. The true value of µ is 75, and Ho is not rejected.
Type II error.
A test is made of Ho: µ = 45 versus Ha: µ < 45. The true value of µ is 40, and Ho is rejected.
Correct decision.
20. State the correct hypotheses:
a.
Last year, the mean amount spent by customers at a certain restaurant was $35. The restaurant
owner believes that the mean may be higher this year.
b.
Ho: µ = $35
Ha: µ > $35
In a recent year, the mean weight of newborn girls in a certain country was 6.6 pounds. A doctor
wants to know whether the mean weight of newborn boys differs from this.
Ho: µ = 6.6 lbs
Ha: µ ≠ 6.6 lbs
The Pew Research Center reported in 2008 that only 15% of 18-24-year-olds read a daily newspaper.
The publisher of a local newspaper wants to know whether the percentage of newspaper readers
among students at a nearby large university differs from the percentage among 18-24-year-olds in
general.
c.
Ho: p = 15%
Ha: p ≠ 15%
A Pew poll taken in 2010 surveyed 830 people aged 18-29, and found that 166 of them had one or
more tattoos. Can you conclude that the percentage of people aged 18-29 who have a tattoo is less
than 22%? (State the hypotheses only.)
Ho: p = 22%
Ha: p < 22%
21. A test is made of Ho: µ = 20 versus Ha: µ ≠ 20. The value of the test statistic is z = -1.84.
a.
For a significance level of α = 0.05, what is (are) the cutoff value(s) of z that get you to the rejection
region?
z = ± 1.96
b.
Find the p-value of the test. Include a sketch. P-value = 0.0658 (Your sketch needs both tails shaded)
c.
Is the data significant at the 5% level?
NO. The p-value is greater than 5%.
22. A test is made of Ho: µ = 5 versus Ha: µ < 5. The value of the test statistic is z = –2.18.
a.
For a significance level of α = 0.05, what is (are) the cutoff value(s) of z that get you to the rejection
region?
z = – 1.645.
b.
Find the p-value of the test. Include a sketch. The p-value is 0.0146.
c.
Do you reject Ho at the 5% level? Yes. The p-value is less than 5%.
Practice Questions for Exam III
Fall 2011
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d.
Is the data significant at the 1% level?
No. The p-value is greater than 1%.
23. A test is made of Ho: µ = 63 versus Ha: µ > 63. The value of the test statistic is z = 2.22.
a.
For a significance level of α = 0.02, what is (are) the cutoff value(s) of z that get you to the rejection
region?
z = 2.054
b.
Find the p-value of the test. Include a sketch. The p-value = 0.0132.
c.
Do you reject Ho at the 2% level? Yes. The p-value is < 2%.
d.
Is the data significant at the 1% level?
No. The p-value is > 1%.
24. The mean height of adult men in the U.S. is 69.7 inches, with a standard deviation of 3 inches. A sociologist
believes that taller men may be more likely to be promoted to positions of leadership, so the mean height µ of
male business executives may be greater than the mean height of the entire male population. A SRS of 100
male business executives has a mean height of 69.9 inches. Assume that the standard deviation of male
business executives is σ = 3 inches. Can we conclude that male business executives are taller, on average, than
the general male population? Use a significance level of α = 0.05.
Ho: µ = 69.7 inches
Z = 0.67
Ha: µ > 69.7 inches
p-value = about 25%
Do not reject Ho.
We cannot conclude that male business executives are taller, on average, than the general male population
(z = 0.67, p-value > 0.05).
25. If the p-value of a test is 0.03, which of the following is the best conclusion?
a.
The probability that Ho is true is 0.03.
b.
The probability that Ho is false is 0.03.
c.
If Ho is true, the probability of obtaining a test statistic as extreme or more extreme as the one
actually observed is 0.03.
d.
If Ho is false, the probability of obtaining a test statistic as extreme as or more extreme than the one
actually observed is 0.03.
26. A certain type of calculator battery has a mean lifetime of 100 hours and a standard deviation of 10 hours. A
company has developed a new battery and claims it has a longer mean life. A random sample of 1000 batteries
is tested, and the sample mean lifetime is 101 hours. The p-value of the test is less than 1%. The battery
manufacturer says that because the evidence is strong that µ > 100, you should be willing to pay a much higher
price for its battery than the old type of battery. Do you agree? Why or why not?
I do NOT agree. Just because the battery lasts “significantly” longer, it lasts only ONE HOUR longer, ON
AVERAGE. This is an increase of only 1%, and that is hardly worth paying “a much higher price” for it.
Statistical significance does not guarantee practical significance.
27. The greater the power of a test, the less likely we are to make a TYPE II error.
28. The power is the probability of rejecting Ho when it is FALSE.
29. If the sample size is increased, the power of the test will INCREASE.
30. A test of Ho: µ = 100 versus Ha: µ > 100 will be made at a significance level of α = 0.01. The population
standard deviation is σ = 25 and the sample size is n = 35. Find the power of the test against the alternative µ a
Practice Questions for Exam III
Fall 2011
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= 110.
The cutoff value of z for the rejection region is z = 2.326 and Ho will be rejected for x-bar values >=
109.829. When the mean is really 110, z = - 0.04 and the power of the test is only 51.6%.
31. You want to test Ho: µ = 4 versus Ha: µ ≠ 4, so you compute a 95% confidence interval for µ. The interval is
(5.1, 7.2).
a.
Do you reject Ho at the 5% significance level? Explain. Yes, because 4 is outside the interval.
b.
Your friend thinks that α = 0.01 is a more appropriate significance level. Can you tell from the CI
whether or not to reject Ho at this level of significance? Explain. No. We know from the fact that we
reject Ho at the 5% level that the p-value of the test is < 5%, but we do not know whether or not it is
less than 1%.
32. The Nielsen Company surveyed 225 owners of Android phones and found that 160 of them planned to get
another Android as their next phone.
a.
Construct a 95% CI estimate for the proportion of Android users who plan to get another Android.
Using the Plus-Four CI Method, the interval is (64.85%, 76.64%), or 70.7% ± 5/9%.
b.
Assume that an advertisement claimed that 70% of Android users plan to get another Android. Does
the confidence interval contradict this claim?
No. 70% is in the range of values the CI predicts for the true proportion.
33. An educator wants to construct a 98% confidence interval for the proportion of elementary school children in
Colorado who are proficient in reading.
a.
The results of a recent statewide test suggested that the proportion is .070. Using this estimate, what
sample size is needed so that the confidence interval will have a margin of error of no more than 0.05?
Use p* = 0.7: n = 455.
b.
Estimate the sample size needed if no estimate of p is available.
Use p* = 0.5: n = 542.
34. A Pew poll taken in 2010 surveyed 830 people aged 18-29, and found that 166 of them had one or more
tattoos. Can you conclude that the percentage of people aged 18-29 who have a tattoo is less than 22%? Carry
out the test. Use a significance level of 4%.
Ho: p = 22%
Ha: p < 22%
z = - 1.39
p-value = 0.0823 > 4%
Do not reject Ho.
We cannot conclude (at the 4% significance level) that the percentage of people aged 18-29 who have a
tattoo is less than 22%.
Practice Questions for Exam III
Fall 2011
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