4. You are hired by the U.S. treasury to determine whether a batch of 5 quarters is counterfeit. Assume you are told that a standard U.S. quarter has a weight that is normally distributed with mean 5.67 grams and standard deviation 0.02 grams. Assume that the weight of counterfeit coins has a distribution with unknown mean and standard deviation. (a) What type of hypothesis test should you use to determine whether the quarters are counterfeit? Be specific. Write down a null and alternative hypothesis for your test. (b) Determine a rejection region for your test at a significance level of alpha = 0.01. (c) Assume you measure the weights of the 5 quarters to be 5.68, 5.65, 5.64, 5.63, and 5.61 respectively. What is the value of your test statistic, and what is its p-value? (d) What inference can you make about the null hypothesis at the alpha = 0.01 significance level?
4. You are hired by the U.S. treasury to determine whether a batch of 5 quarters is counterfeit. Assume you are told that a standard U.S. quarter has a weight that is normally distributed with mean 5.67 grams and standard deviation 0.02 grams. Assume that the weight of counterfeit coins has a distribution with unknown mean and standard deviation. (a) What type of hypothesis test should you use to determine whether the quarters are counterfeit? Be specific. Write down a null and alternative hypothesis for your test. (b) Determine a rejection region for your test at a significance level of alpha = 0.01. (c) Assume you measure the weights of the 5 quarters to be 5.68, 5.65, 5.64, 5.63, and 5.61 respectively. What is the value of your test statistic, and what is its p-value? (d) What inference can you make about the null hypothesis at the alpha = 0.01 significance level?
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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![**Educational Exercise on Statistical Hypothesis Testing**
**Problem Statement:**
You are hired by the U.S. Treasury to determine whether a batch of 5 quarters is counterfeit. Assume you are told that a standard U.S. quarter has a weight that is normally distributed with a mean of 5.67 grams and a standard deviation of 0.02 grams. Assume that the weight of counterfeit coins has a distribution with unknown mean and standard deviation.
**Questions:**
**(a)** What type of hypothesis test should you use to determine whether the quarters are counterfeit? Be specific. Write down a null and alternative hypothesis for your test.
**Answer:**
Use a one-sample t-test to compare the sample mean against the population mean. The null hypothesis (H0) would be that the mean weight of the quarters is 5.67 grams (not counterfeit). The alternative hypothesis (H1) would be that the mean weight of the quarters is not 5.67 grams (counterfeit).
**(b)** Determine a rejection region for your test at a significance level of alpha = 0.01.
**Answer:**
For a two-tailed test with α = 0.01, check the t-distribution table for critical t-values corresponding to 4 degrees of freedom. The rejection region is the range of values less than the negative critical value or greater than the positive critical value.
**(c)** Assume you measure the weights of the 5 quarters to be 5.68, 5.65, 5.64, 5.63, and 5.61 grams, respectively. What is the value of your test statistic, and what is its p-value?
**Answer:**
Calculate the mean and standard deviation of the sample weights. Use the t-formula:
\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]
where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size. Compute the t-value and determine the p-value from the t-distribution with 4 degrees of freedom.
**(d)** What inference can you make about the null hypothesis at the alpha = 0.01 significance level?
**Answer:**
Compare the calculated p-value with the significance level (α = 0.01).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F358deae0-b7d2-45f9-8056-32e7a5350251%2Fd44b3b5d-86f1-49ae-b5c7-572c2e0b717c%2Fd0bnlgn_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Exercise on Statistical Hypothesis Testing**
**Problem Statement:**
You are hired by the U.S. Treasury to determine whether a batch of 5 quarters is counterfeit. Assume you are told that a standard U.S. quarter has a weight that is normally distributed with a mean of 5.67 grams and a standard deviation of 0.02 grams. Assume that the weight of counterfeit coins has a distribution with unknown mean and standard deviation.
**Questions:**
**(a)** What type of hypothesis test should you use to determine whether the quarters are counterfeit? Be specific. Write down a null and alternative hypothesis for your test.
**Answer:**
Use a one-sample t-test to compare the sample mean against the population mean. The null hypothesis (H0) would be that the mean weight of the quarters is 5.67 grams (not counterfeit). The alternative hypothesis (H1) would be that the mean weight of the quarters is not 5.67 grams (counterfeit).
**(b)** Determine a rejection region for your test at a significance level of alpha = 0.01.
**Answer:**
For a two-tailed test with α = 0.01, check the t-distribution table for critical t-values corresponding to 4 degrees of freedom. The rejection region is the range of values less than the negative critical value or greater than the positive critical value.
**(c)** Assume you measure the weights of the 5 quarters to be 5.68, 5.65, 5.64, 5.63, and 5.61 grams, respectively. What is the value of your test statistic, and what is its p-value?
**Answer:**
Calculate the mean and standard deviation of the sample weights. Use the t-formula:
\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]
where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size. Compute the t-value and determine the p-value from the t-distribution with 4 degrees of freedom.
**(d)** What inference can you make about the null hypothesis at the alpha = 0.01 significance level?
**Answer:**
Compare the calculated p-value with the significance level (α = 0.01).
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