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Chapter 9: Problem 46

A company claims that the mean net weight of the contents of its All Taste cereal boxes is at least 18 ounces. Suppose you want to test whether or not the claim of the company is true. Explain briefly how you would conduct this test using a large sample. Assume that \(\sigma=.25\) ounce.

Short Answer

Expert verified
We would conduct a one-sample z-test for the hypothesis if the mean weight of the cereal boxes is indeed 18 ounces. However, this step-by-step doesn't lead to a concrete conclusion as we lack the sample mean and size for calculation.

Step by step solution

01

State the Hypotheses

In this case, our null hypothesis (H0) is that the mean net weight of the cereal boxes is 18 ounces. This can be written in statistical terms as \(H0: \mu = 18\). The alternative hypothesis (Ha), which is what we're testing, is if the mean net weight is less than 18 ounces. In statistical terms, this can be written as \(Ha: \mu < 18\) ounces.
02

Calculate the Standard Error

We know that \(\sigma = .25\) ounce, and the sample size (n) is large. The standard error (SE) is given by \(\sigma/\sqrt{n}\). Without a provided sample size, we'll use 'n' in the equation for now.
03

Conduct the Z-Test

The test statistic for a z-test is calculated as \(Z= (X - \mu) / SE\) where X is the sample mean. In our case, we don't have a sample mean, so we'll use 'X' in the equation for now.
04

Find the p-value

After calculating the Z-score, refer to a Z-table or use a statistical calculator to find the p-value associated with that Z-score.
05

Make a Conclusion

If the p-value is less than the chosen significance level (usually 0.05), reject the null hypothesis. This would indicate that the mean weight of the cereal boxes is less than 18 ounces, which would contradict the company's claim. If the p-value is higher than the significance level, do not reject the null hypothesis. The result isn't convincing enough to say the mean weight isn't 18 ounces.

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Most popular questions from this chapter

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=45, \quad H_{1}: \mu>45\) b. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23\) c. \(H_{0}: \mu \geq 75, \quad H_{1}: \mu<75\)

The past records of a supermarket show that its customers spend an average of \(\$ 95\) per visit at this store. Recently the management of the store initiated a promotional campaign according to which each customer receives points based on the total money spent at the store, and these points can be used to buy products at the store. The management expects that as a result of this campaign, the customers should be encouraged to spend more money at the store. To check whether this is true, the manager of the store took a sample of 14 customers who visited the store. The following data give the money (in dollars) spent by these customers at this supermarket during their visits. \(\begin{array}{rrrrrrr}109.15 & 136.01 & 107.02 & 116.15 & 101.53 & 109.29 & 110.79 \\ 94.83 & 100.91 & 97.94 & 104.30 & 83.54 & 67.59 & 120.44\end{array}\) Assume that the money spent by all customers at this supermarket has a normal distribution. Using the \(5 \%\) significance level, can you conclude that the mean amount of money spent by all customers at this supermarket after the campaign was started is more than \(\$ 95 ?\) (Hint: First calculate the sample mean and the sample standard deviation for these data using the formulas learned in Sections \(3.1 .1\) and \(3.2 .2\) of Chapter \(3 .\) Then make the test of hypothesis about \(\mu .\) )

Records in a three-county area show that in the last few years, Girl Scouts sell an average of \(47.93\) boxes of cookies per year, with a population standard deviation of \(8.45\) boxes per year. Fifty randomly selected Girl Scouts from the region sold an average of \(46.54\) boxes this year. Scout leaders are concerned that the demand for Girl Scout cookies may have decreased. a. Test at the \(10 \%\) significance level whether the average number of boxes of cookies sold by all Girl Scouts in the three-county area is lower than the historical average. b. What will your decision be in part a if the probability of a Type I error is zero? Explain.

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=120 \text { versus } H_{1}: \mu>120 $$ A random sample of 81 observations taken from this population produced a sample mean of \(123.5 .\) The population standard deviation is known to be 15 . a. If this test is made at the \(2.5 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01 ?\) What if \(\alpha=.05\) ?

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.
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