We continue to study confidence intervals for a population mean, but this time, we will not assume the population SD is known so we will use the standard deviation obtained or calculated from the sample (and we will use the t-distribution). Note: When the population standard deviation is unknown and the parameter of interest is the mean, we use the t-distribution. Recall that confidence intervals are a range of values used to estimate a population parameter and are associated with a level of confidence. If the data is quantitative, we will be estimating the population mean, but if the data is qualitative, we will estimate the population proportion or percentage.
To Do
Watch the videos and use the guided notes
Read chapter 7.2 pages 329 -339. If you understand the guided notes, you can skip the reading.
Complete the Confidence Interval Problem below and upload your work. Please write legibly.
Video 1: We will be using the t-distribution when the population standard deviation is unknown. Here is a video with a little bit of history on the t-distribution.
t-distribution
Video 2: Ch 7.2 Part I. In this video, you will learn to construct confidence intervals for estimating the population mean when the population standard deviation is unknown. Since the Population SD is unknown, we will use the t-distribution. To find the t-score use invT, not the table. And the area is from the left.
Find the t-score using the Calculator –> 2nd then distribution choose InvT enter the cumulative area from the left and then the degrees of freedom
Example: 95% level of confidence with a sample size of 15. Calculator: InvT(0.025,14) or invT(0.975,14)
Ch 7.2 Part I
Video 3: Ch 7.2 Part II. More examples, however, these examples involve raw data. You will also learn about sample size.
Ch 7.2 Part II
Note: to find the t-score, use InvT(Area,df) where Area is the area from the left and degrees of freedom (df) is n-1.
Problem:
A researcher wants to estimate the average number of hours college students sleep per night. They randomly sample 15 students and record their sleep hours for a week. The sample mean is 6.8 hours with a sample standard deviation of 1.2 hours. Assume that the data follow a normal distribution.
Using this sample, construct a 95% confidence interval for the population mean sleep time of college students. Since the population standard deviation is unknown and the sample size is small, use the t-distribution.
Questions:
Why is the t-distribution used instead of the z-distribution in this case?
What does the 95% confidence interval tell us about the true population mean?