Homework Assignment 4

Stat 226 Homework4 Fall 2022 All Sections Submission: via Canvas This homework project focuses on the sampling distribution of the sample mean and the Central Limit Theorem (CLT) . Instructions: This homework assignment consists of two parts: • Part 1: Automated grading/numerical response assessment (3 attempts, highest score kept out of 25 points) • Part 2: Free response assessment (1 attempt, graded offline). There is one free response question for this assignment, and it is extra credit (worth 1 point). Before beginning with the assignment, here is a quick summary of the Central Limit Theorem (CLT) and why it is useful. We are generally interested in knowing something about the population parameters µ and σ , which are typically unknown. But we can use the CLT, which says the following: Suppose that some variable of interest in a population has a distribution with mean µ and standard deviation σ . If we draw a simple random sample of size n from the population and compute the sample mean X ¯ of the variable, then • The sample mean X ¯ follows approximately a Normal distribution. [This is true even if the distribution of the variable in the original population is not Normal as long as the sample size is large enough!]. • The mean of the sampling distribution of X ¯ equals the population mean µ . We write this as follows: µ X ¯ = µ . • The standard deviation of the sampling distribution of√ X ¯ equals the population standard devi- ation σ divided by the square root of the sample size n . We write this as follows: . In summary, the CLT says that if a variable X follows some distribution with mean µ and standard deviation σ , regardless of the shape of the distribution (i.e., whether the distribution is Normal or√ not), then the sample mean is normally distributed: X ¯ N( ∼ µ, / n σ ). We use SE( X ¯ ) to denote σ X ¯ . That is, SE( . So, why is the sampling distribution of the sample mean and the CLT useful in order to learn something about the unknown population mean µ ? 1

The answer is that, when we take a sample from the population and calculate the corresponding sample mean, we can use this sample mean as an estimate for µ . Our sample mean is considered a good estimate because the sample means vary around the unknown population mean µ (bullet point 2 above) and we can reduce the variation by which the sample means vary about µ by increasing the sample size (bullet point 3). Hence, even though our sample mean is not equal to the unknown population mean µ , we can expect it to be very close to µ . How close we want ¯ x to be to µ can be controlled by the sample size and standard deviation σ . The assignment begins on the following page. 1. Describing the Sampling Distribution of X ¯ . A manufacturer of automobile batteries claims that the length of life for their batteries is normally distributed with a mean of 66 months and a standard deviation of 9 months (i.e. battery life N(66 ∼ , 9)). (a) Describe the sampling distribution of the sample mean for samples of size 16, i.e. describe the distribution of the average life time of these automobile batteries for samples of size 16. i. Shape (Choose one): • Normal • Approximately Normal • Not Normal ii. Mean: 66 iii. Standard Error: 2.25 (Do NOT round answer.) (b) How would your answer in part (a) change if the battery life follows a skewed left distribution? i. Shape (Choose one): • Normal • Approximately Normal • Not Normal ii. Mean: 66 iii. Standard Error: 2.25 (Do NOT round answer.) (c) How would your answer in part (a) change if the battery life follows a skewed left distribution and the sample size was 81 instead of 16? i. Shape (Choose one): • Normal 2