This covers sections 8.1, 8.2, and 8.3. In section 8.3, we will not be covering how to deal with
Small Samples. We will NOT cover 8.4. 8.5 is a review, so it’s worth reading. Next week, we will
move on to chapter 9.
Homework:
Section 8.1: 43, 44, 49, 50
Section 8.2: 29, 30, 35, 36
Section 8.3: 27, 28
1)
A sample of size n = 80 is drawn from a population whose standard deviation is σ = 6.8. The
sample mean is 𝑥̅ = 40.4.
Construct a 90% confidence interval for µ
Does the population need to be distributed normally for the confidence interval to be valid?
2) Look at the z-interval menu on the calculator. Look at the t-interval menu on the calculator.
How are they different (using either Stats or Data mode)?
So what is the difference between a Z-interval and a t-interval?
-----------------------A study is done to discover the average amount of money that households donate to non-profits.
Assumption: distribution of donations is approximately normal (our sample size is not >30) so the
distribution of the mean is approximately normal.
After surveying 25 households, a sample mean of $732 was found. The population
standard deviation is known to be $120.
Using the calculator, construct a 95% confidence interval for the mean donation.
What’s the standard error for the sample mean?
What’s the critical zα/2 for a 95% confidence interval?
What’s the margin of error for a 95% confidence interval?
We would like to have a margin of error be +/- of $10 (at 95% confidence). How many
household would need to be sampled to have a margin of error be $10?
z
n /2
m
2
Problem is same as above, but population standard deviation is not known
Assumption: distribution of donations is approximately normal (our sample size is not >30) so the
distribution of the mean is approximately normal.
After surveying 25 households, a sample mean of $732 was found. The standard deviation
of the sample is $120
Since we don’t have the population standard deviation and must use a sample, we will not
use a normal distribution and z-scores. Instead we will use which distribution?
Construct a 95% confidence interval for the mean donation.
How many degrees of freedom are there?
What’s the estimated standard error for the sample mean?
Arguments for the inverted t-distribution is invT(area in left tail, degrees freedom)
What’s the critical tα/2 for a 95% confidence interval?
What’s the margin of error for a 95% confidence interval?
Note: when sample size gets large (>200), the t-distribution becomes very close to the normal
distribution, and some people will just switch to normal for critical values. However, one should still
refer to the distribution as a t-distribution if the population standard deviation is not known.
--------------------------Proportions:
A survey of 150 adults found that 38% of those surveyed bought music over the internet.
What are the conditions necessary to construct a confidence interval for a population
proportion?
Construct a 98% confidence interval for the population mean.
What’s the standard error for the proportion?
What’s the critical zα/2 for a 98% confidence interval?
What’s the margin of error for a 98% confidence interval?
If the music industry wants to conduct another survey, and they want the margin of error to
be 1% point, how many adults will need to be surveyed (still 98% confidence)?
z
If there’s an estimate for p̂ : n pˆ (1 pˆ ) / 2
E
2
z
If not: n .25 / 2
E
2
0
