Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.T.3

Chapter 7, Problem 7.T.3

A travel analyst says that the mean price of a meal for a family of 4 in a resort restaurant is at most \$100. A random sample of 33 meal prices for families of 4 has a mean of \$110 and a standard deviation of \$19. At α=0.01, is there enough evidence to reject the analyst’s claim?

Verified step by step guidance

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ ≤ 100 (the mean price is at most $100), and the alternative hypothesis is H₁: μ > 100 (the mean price is greater than $100). This is a one-tailed test.

Step 2: Determine the test statistic formula for a one-sample t-test. The formula is t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized mean, s is the sample standard deviation, and n is the sample size.

Step 3: Substitute the given values into the formula. Here, x̄ = 110, μ₀ = 100, s = 19, and n = 33. Calculate the test statistic t using these values.

Step 4: Determine the critical t-value for a one-tailed test at α = 0.01 with degrees of freedom df = n - 1 = 33 - 1 = 32. Use a t-distribution table or statistical software to find the critical t-value.

Step 5: Compare the calculated t-value to the critical t-value. If the calculated t-value is greater than the critical t-value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis states that the mean price of a meal is at most $100, while the alternative suggests it is greater. The goal is to determine if the sample data provides sufficient evidence to reject the null hypothesis.

P-Value

The p-value is a measure that helps determine the significance of the results in hypothesis testing. It represents the probability of observing the sample data, or something more extreme, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis. In this scenario, the p-value will be compared to the significance level (α = 0.01) to decide whether to reject the analyst's claim.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. It provides an estimate of the uncertainty around the sample mean. In this case, constructing a confidence interval for the mean meal price can help assess whether the true mean could be above $100, thus supporting or refuting the analyst's claim.

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Related Practice

Textbook Question

A nutrition bar manufacturer claims that the standard deviation of the number of grams of carbohydrates in a bar is 1.11 grams. A random sample of 26 bars has a standard deviation of 1.19 grams. At α=0.05, is there enough evidence to reject the manufacturer’s claim? Assume the population is normally distributed.

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Textbook Question

In Exercises 7–10, explain how you should interpret a decision that fails to reject the null hypothesis.

A nonprofit consumer organization says that the standard deviation of the starting prices of its top-rated vehicles for a recent year is no more than \$2900.

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Textbook Question

[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

370

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Textbook Question

You want your test to support a positive claim about your college, not just fail to reject one. Should you state your claim so that the null hypothesis contains the claim or the alternate hypothesis contains the claim? Explain.

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Textbook Question

When you reject a true claim with a level of significance that is virtually zero, what can you infer about the randomness of your sampling process?

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Textbook Question

n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.

σ > 1.9

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