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Chapter 9: Problem 36

Tomatoes The weights of four randomly and independently selected bags of tomatoes labeled 5 pounds were found to be \(5.1\), \(5.0,5.3\), and \(5.1\) pounds. Assume Normality. a. Find a \(95 \%\) confidence interval for the mean weight of all bags of tomatoes. b. Does the interval capture \(5.0\) pounds? Is there enough evidence to reject a mean weight of \(5.0\) pounds?

Short Answer

Expert verified
The 95% confidence interval for the mean weight of all bags of tomatoes is determined by using the earlier computed values of mean, standard error and the Z-value. The determination of whether this interval captures 5.0 pounds will be based on whether 5.0 is contained within this confidence interval.

Step by step solution

01

Determine the mean of given weights

Add the 4 given weights and divide by 4 to calculate the mean. Mean \( \mu = \frac{5.1 + 5.0 + 5.3 + 5.1}{4} \)
02

Determine the standard deviation (SD)

First, find the differences between each given weight and the mean weight, square these differences, add them together and divide by 4-1=3 (because we apply Bessel's correction for sample SD). Take a square root of the result to find the standard deviation. SD \( = \sqrt{\frac{(5.1-\mu)^2 + (5.0-\mu)^2 + (5.3-\mu)^2 + (5.1-\mu)^2}{3}} \)
03

Determine the standard error (SE)

SE = SD/ \(\sqrt{n}\), where n = the number of bags of tomatoes = 4.
04

Calculate the 95% confidence interval for the mean weight

Using the formula for a 95% confidence interval for the mean, we find the interval by using \(\mu \pm Z(SE)\) , where Z is the Z-statistic for the 95% confidence level = 1.96. This provides upper and lower limits for the mean weight with a 95% confidence interval.
05

Verify if the interval contains 5.0 pounds

Check if 5.0 is between the confidence limits found in step 4.

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Most popular questions from this chapter

Oranges A statistics instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags. They weighed \(10.2,10.5,10.3\), and \(10.3\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of oranges. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. I am \(95 \%\) confident that the population mean is between ii. There is a \(95 \%\) chance that all intervals will be between iii. I am \(95 \%\) confident that the sample mean is between b. Does the interval capture 10 pounds? Is there enough evidence to reject the null hypothesis that the population mean weight is 10 pounds? Explain your answer.

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