Statistics6.2: Confidence Intervals for the Mean (small samples)
Objective 1: I can interpret the t-distribution and use a t-distribution table.
In most real-life situations, the population standard deviation is unknown. And often, it is not practical to collect samples of 30 and more. So, how can we construct a confidence interval for a population mean given such circumstances? If the random variable is normally distributed, you can use a ______.
Properties of the t-distribution:
1) The critical values of the t-distribution are denoted by
2) The t-distribution is
3) The t-distribution is a ______, each determined by a parameter called the ______. The ______(d.f.) = ______.
4) The total area under a t-curve is ____ or ______.
5) The mean, median, and mode of the t-distribution are equal to ______.
6) As the ______increase, the t-distribution approaches the ______. After 30 d.f., the t-distribution is very close to the ______.
Table 5 on pg A18 (or on the foldout chart) is used to find the critical values for the t-distribution.
Example 1: Find the critical value, ____, for a 95% confidence when the sample size is 15.
*This means that 95% of the area under the t-curve falls between _____ and ______.
TIY 1: Find the critical value ____ for a 90% confidence when the sample size is 22.
*This means that 90% of the area under the t-curve falls between _____ and ______.
Objective 2: I can construct a confidence interval using the t-distribution.
Constructing a CI using the t-distribution is similar to constructing a CI using the normal distribution (section 6.1). Both use a point estimate (x) and a margin of error, E.
Steps to constructing a CI for the t-distribution by hand:
1) Identify the needed sample statistics,2) Identify degrees of freedom, level of confidence, and the critical value, ____.
3) Calculate the margin of error, E.
4) Find the left and right endpoints and form the ______.
Example 2: You randomly select 16 coffee shops and measure the temperature of the coffee sold at each shop. The sample mean temperature is 162.0 with a sample standard deviation of 10.0. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed.
Wanna see the calculator shortcut????
TIY 2: Find the 90% and 99% confidence intervals using the information in Example 2. Compare all 3 confidence intervals. What do you notice?
Example 3: You randomly select 20 mortgage companies and determine the current mortgage interest rate at each. The sample mean rate is 6.22% with a sample standard deviation of 0.42%. Find the 99% confidence interval for the population mean mortgage interest rate. Assume the interest rates are approximately normally distributed.
TIY 3: Find the 90% and 95% confidence intervals using the information in Example 3. What do you notice?
So how do you know when to use a z-interval or t-interval? Check out the handy flowchart on page 329 in your book. It’s too large to reproduce in your notes, but here are the general rules.
Use the z-interval when
1)
2)
Use the t-interval when
Example 4: You randomly select 25 newly constructed homes. The sample mean cost is $181,000 with a population standard deviation of $28,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction costs? Explain your reasoning.
TIY 4: You randomly select 18 male athletes and measure their resting heart rate. The sample mean heart rate is 64 beats/minute with a sample SD of 2.5 beats/min. Assuming heart rates are normally distributed, should you use the normal distribution or the t-distribution, or neither to construct a 90% confidence interval for the mean heart rate? Explain your reasoning.