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Chapter 9: Problem 84

French Fries A fast-food chain advertises that the mediumsize serving of French fries weighs 135 grams. A reporter took a random sample of 10 medium orders of French fries and weighed each order. The weights (in grams) were \(111,124,125,156,127,134\),\(135,136,139,141 .\) Assume the population distribution is Normal. (Source: soranews24.com) a. Test the hypothesis that the medium servings have a population mean different from 135 grams. Use a significance level of \(0.05\). b. Construct a \(95 \%\) confidence interval for the population mean. How does your confidence interval support your conclusion in part a? Do you think the consumers are being misled about the serving size? Explain.

Short Answer

Expert verified
To find the short answer, check the result of the hypothesis test and the confidence interval created. If the null hypothesis was not rejected and the interval includes 135g, the claim by the fast-food chain is plausible. If not, they might be misleading the consumers.

Step by step solution

01

Calculate sample mean and sample standard deviation

Firstly gather the provided measurements: \(111,124,125,156,127,134,135,136,139,141\) and calculate the sample mean (\(\overline X\)) and sample standard deviation (s). The sample mean is simply the average of your data, and the standard deviation measures how spread out the numbers in the data are.
02

Formulate the null and alternate hypotheses

The null hypothesis (\(H_0\)) is that the medium servings have a population mean equal to 135 grams (\(u = 135\)). The alternate hypothesis (\(H_a\)) is that the medium servings have a population mean different from 135 grams (\(u \neq 135\)).
03

Conduct a t-test and compare the test statistic to a critical value

Conducting the t-test involves calculating the test statistic, known as the t-value. This is given by the formula \(t = \frac{(\overline X - u)}{s/\sqrt{n}}\) where \(n\) is the sample size. The t-value compares the observed difference between the sample mean and the hypothesised population mean, to the variation expected due to sampling error. The critical value can be found from the t-distribution table for a two-tailed test at the significance level of 0.05 and degrees of freedom \(n-1\). If the absolute value of the calculated t-value is greater than the t critical value, we reject the null hypothesis.
04

Construct a 95% confidence interval for the population mean

Using the formula \(\overline X \pm (t_{critical} \cdot \frac{s}{\sqrt{n}})\), we can create a 95% confidence interval for the population mean. The value of \(t_{critical}\) is the same as the one from the t-distribution table used in step 3.
05

Compare the confidence interval with the claimed serving size

Verify if the confidence interval includes 135 grams. If it does include 135g, the claim by the fast-food chain is at least plausible given the collected data. If it doesn't, we can assume the consumers are being misled about the serving size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null and Alternate Hypotheses
When it comes to hypothesis testing, the foundational step is establishing the null and alternate hypotheses. The null hypothesis, denoted as \( H_0 \), represents a statement of no effect or no difference 鈥� it's the assumption that the status quo, or the claim being tested, is true. For example, if a fast-food chain claims their serving size is 135 grams, the null hypothesis would assert that the true population mean serving size is indeed 135 grams, symbolized mathematically as \( H_0: \mu = 135 \) grams.

Conversely, the alternate hypothesis, denoted as \( H_a \) or \( H_1 \), suggests that there is an effect or a difference. It contests the claim of the null hypothesis. In the given exercise, the alternate hypothesis posits that the true population mean serving size is different from 135 grams, expressed as \( H_a: \mu eq 135 \) grams. The choice of the alternate hypothesis implies the direction of the test. Since \

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Most popular questions from this chapter

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