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According to the Current Results website, the state of California has a mean annual rainfall of 21 inches, whereas the state of New York has a mean annual rainfall of 38 inches. Assume that the standard deviation for both states is 3 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. Use z-table. a. Show the probability distribution of the sample mean annual rainfall for California. and σ = (to 4 decimals). This is a normal probability distribution with E() b. What is the probability that the sample mean is within 1 inch of the population mean for California? (to 4 decimals) c. What is the probability that the sample mean is within 1 inch of the population mean for New York? (to 4 decimals) d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why? The probability of being within 1 inch is greater for New York in part (c) because the sample size is larger

According to the Current Results website, the state of California has a mean annual rainfall of 21 inches, whereas the state of New York has a mean annual rainfall of 38 inches. Assume that the standard deviation for both states is 3 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. Use z-table. a. Show the probability distribution of the sample mean annual rainfall for California. and σ = (to 4 decimals). This is a normal probability distribution with E() b. What is the probability that the sample mean is within 1 inch of the population mean for California? (to 4 decimals) c. What is the probability that the sample mean is within 1 inch of the population mean for New York? (to 4 decimals) d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why? The probability of being within 1 inch is greater for New York in part (c) because the sample size is larger

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Understanding Probability Distribution and Sample Mean Analysis #### Context According to the Current Results website, the state of California has a mean annual rainfall of 21 inches, whereas the state of New York has a mean annual rainfall of 38 inches. The standard deviation for both states is 3 inches. We analyze a sample of 30 years of rainfall for California and a sample of 45 years for New York using a z-table. #### Analysis **a. Probability Distribution for California's Sample Mean** - We use a normal probability distribution. - The expected value of the sample mean, \( E(\bar{x}) \), is required. - The standard error, \( \sigma_{\bar{x}} \), needs to be calculated to four decimal places. **b. Probability for California's Sample Mean** - Determine the probability that the sample mean is within 1 inch of the mean for California. - Use the z-table for precise calculation to four decimal places. **c. Probability for New York's Sample Mean** - Find the probability that the sample mean is within 1 inch of the mean for New York. - Calculate to four decimal places using the z-table. **d. Comparability of Probabilities** - Compare probabilities from parts (b) and (c) to determine where the probability of being within 1 inch of the population mean is greater. - The analysis concludes that the probability is greater for New York in part (c) because the sample size is larger. This structured analysis helps us understand the impact of sample size and standard deviation on the probability of sample means being within a certain range of the population mean.
Transcribed Image Text:### Understanding Probability Distribution and Sample Mean Analysis #### Context According to the Current Results website, the state of California has a mean annual rainfall of 21 inches, whereas the state of New York has a mean annual rainfall of 38 inches. The standard deviation for both states is 3 inches. We analyze a sample of 30 years of rainfall for California and a sample of 45 years for New York using a z-table. #### Analysis **a. Probability Distribution for California's Sample Mean** - We use a normal probability distribution. - The expected value of the sample mean, \( E(\bar{x}) \), is required. - The standard error, \( \sigma_{\bar{x}} \), needs to be calculated to four decimal places. **b. Probability for California's Sample Mean** - Determine the probability that the sample mean is within 1 inch of the mean for California. - Use the z-table for precise calculation to four decimal places. **c. Probability for New York's Sample Mean** - Find the probability that the sample mean is within 1 inch of the mean for New York. - Calculate to four decimal places using the z-table. **d. Comparability of Probabilities** - Compare probabilities from parts (b) and (c) to determine where the probability of being within 1 inch of the population mean is greater. - The analysis concludes that the probability is greater for New York in part (c) because the sample size is larger. This structured analysis helps us understand the impact of sample size and standard deviation on the probability of sample means being within a certain range of the population mean.
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