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Chapter 10: Problem 38

The article "Theaters Losing Out to Living Rooms" (San Luis Obispo Tribune, June 17,2005\()\) states that movie attendance declined in \(2005 .\) The Associated Press found that 730 of 1000 randomly selected adult Americans preferred to watch movies at home rather than at a movie theater. Is there convincing evidence that the majority of adult Americans prefer to watch movies at home? Test the relevant hypotheses using a .05 significance level.

Short Answer

Expert verified
Yes, there is convincing evidence at the 0.05 significance level that the majority of adult Americans prefer to watch movies at home over going to a movie theater.

Step by step solution

01

Formulate Hypotheses

Let \( p \) be the proportion of all adults who prefer to watch movies at home. The null hypothesis \(H_0\) is that the majority of adult Americans do not prefer to watch movies at home, so \( p \leq 0.5 \). The alternative hypothesis \(H_1\) is that the majority do prefer to watch movies at home, so \( p > 0.5 \).
02

Compute Test Statistic

For a sample of size \(n = 1000\), the sample proportion \(\hat{p} = 730/1000 = 0.73\). The test statistic is computed as \( Z = (\hat{p} - p_0) / \sqrt{(p_0*(1-p_0)/n)} \), where \(p_0 = 0.5\) under the null hypothesis. Substituting values, we have: \( Z = (0.73 - 0.5) / \sqrt{(0.5*(1-0.5)/1000)} = 9.2 \).
03

Acceptance / Rejection of Null Hypothesis

As we're dealing with a one-tailed test with significance level of 0.05, we have to calculate the critical Z value. For 0.05 in the one-tailed test, the critical value is approximately 1.645. As our test statistic (\(Z = 9.2\)) is greater than the critical value (\(Z = 1.645\)), we reject the null hypothesis.
04

Interpretation

Thus, there is strong evidence at the 0.05 significance level to suggest that the majority of adult Americans do prefer to watch movies at home over going to a movie theater.

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Key Concepts

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

Significance Level
In hypothesis testing, the significance level is crucial. It's a threshold we set for how much error we're willing to accept when deciding whether to reject the null hypothesis.
In simpler terms, it's the probability of making a mistake by rejecting a true null hypothesis.
  • Typically, a significance level is denoted by the symbol \( \alpha \).
  • Common choices for \( \alpha \) are 0.05, 0.01, and 0.10.
  • A 0.05 significance level means there's a 5% chance you'll mistakenly reject the null hypothesis.
In our example, the significance level is set at 0.05. This means we have a standard 5% risk of ruling out the null hypothesis incorrectly. By setting this level, we decide how strict we will be with our evidence, ensuring that findings are robust and reliable.
Null Hypothesis
The null hypothesis, often symbolized as \( H_0 \), is the default assumption in hypothesis testing.
It asserts that any observed difference is due to chance alone and there's no effect or relationship.
  • The purpose of \( H_0 \) is to provide a baseline or standard to compare against.
  • In our movie preference study, \( H_0 \) states that the majority of adult Americans do not prefer to watch movies at home, mathematically represented as \( p \leq 0.5 \).
  • We always test \( H_0 \) to see if there's enough evidence to refute it.
The null hypothesis acts as a skeptic, assuming the status quo until we provide convincing data suggesting otherwise.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), proposes a new state of affairs, in contrast to the null hypothesis.
It's what researchers aim to support with their data.
  • \( H_1 \) is typically set up to capture the effect or relationship under investigation.
  • In our scenario, \( H_1 \) suggests that the majority of adult Americans do prefer to watch movies at home, expressed as \( p > 0.5 \).
  • The acceptance of \( H_1 \) implies that the evidence significantly contradicts \( H_0 \).
If our testing leads to rejecting the null hypothesis, we're essentially accepting the alternative hypothesis as a more accurate reflection of reality.
Test Statistic
A test statistic is a standardized value that helps decide whether to reject the null hypothesis.
It measures how far the sample data deviates from the null hypothesis assumption.
  • In many cases, the test statistic follows a known distribution, such as the normal distribution.
  • For our study, a Z-test is used, resulting in a Z statistic of the form \( Z = (\hat{p} - p_0) / \sqrt{(p_0(1 - p_0)/n)} \).
  • This formula incorporates the difference between the sample proportion \( \hat{p} = 0.73 \) and the null hypothesis proportion \( p_0 = 0.5 \).
  • The calculated Z statistic is 9.2, which tells us how many standard deviations our sample proportion is from the population proportion under \( H_0 \).
The test statistic provides a way to evaluate whether the observed data is significantly different from what we would expect under the null hypothesis.
Critical Value
In hypothesis testing, the critical value is a threshold that the test statistic must exceed to reject the null hypothesis.
It defines the boundary for the likelihood of data corresponding to the null hypothesis.
  • Critical values are derived from the chosen significance level and the statistical distribution of the test statistic.
  • In a normal distribution, the critical value can be looked up in Z-tables when conducting a Z-test.
  • Here, for a significance level of 0.05 in a one-tailed test, the critical value is approximately 1.645.
  • The test statistic (9.2) significantly exceeds this critical value, prompting the rejection of the null hypothesis.
Therefore, when the test statistic surpasses the critical value, it indicates that the sample data is inconsistent with the null hypothesis, supporting the alternative hypothesis instead.

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Most popular questions from this chapter

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

For the following pairs, indicate which do not comply with the rules for setting up hypotheses, and explain why: a. \(H_{0}: \mu=15, H_{a}: \mu=15\) b. \(H_{0}: p=.4, H_{a}: p>.6\) c. \(H_{0}: \mu=123, H_{a}: \mu<123\) d. \(H_{0}: \mu=123, H_{d}: \mu=125\) e. \(\quad H_{0}: \hat{p}=.1, H_{a}: \hat{p} \neq .1\)

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), a scientist will take 50 water samples at randomly selected times and will record the water temperature of each sample. She will then use a \(z\) statistic $$ z=\frac{\bar{x}-150}{\frac{\sigma}{\sqrt{n}}} $$ to decide between the hypotheses \(H_{0}: \mu=150\) and \(H_{a}: \mu>150,\) where \(\mu\) is the mean temperature of discharged water. Assume that \(\sigma\) is known to be 10 . a. Explain why use of the \(z\) statistic is appropriate in this setting. b. Describe Type I and Type II errors in this context. \(c\). The rejection of \(H_{0}\) when \(z \geq 1.8\) corresponds to what value of \(\alpha\) ? (That is, what is the area under the \(z\) curve to the right of \(1.8 ?\) ) d. Suppose that the actual value for \(\mu\) is 153 and that \(H_{0}\) is to be rejected if \(z \geq 1.8 .\) Draw a sketch (similar to that of Figure 10.5 ) of the sampling distribution of \(\bar{x},\) and shade the region that would represent \(\beta\), the probability of making a Type II error. e. For the hypotheses and test procedure described, compute the value of \(\beta\) when \(\mu=153\). f. For the hypotheses and test procedure described, what is the value of \(\beta\) if \(\mu=160\) ? g. What would be the conclusion of the test if \(H_{0}\) is rejected when \(z \geq 1.8\) and \(\bar{x}=152.4\) ? What type of error might have been made in reaching this conclusion?

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of \(n=12\) frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations, along with a boxplot and normal probability plot: \(\begin{array}{llllllll}255 & 244 & 239 & 242 & 265 & 245 & 259 & 248\end{array}\) \(\begin{array}{llll}225 & 226 & 251 & 233\end{array}\) a. Is it reasonable to test hypotheses about mean calorie content \(\mu\) by using a \(t\) test? Explain why or why not. b. The stated calorie content is \(240 .\) Does the boxplot suggest that true average content differs from the stated value? Explain your reasoning. c. Carry out a formal test of the hypotheses suggested in Part (b).

10.52 - Medical research has shown that repeated wrist extension beyond 20 degrees increases the risk of wrist and hand injuries. Each of 24 students at Cornell University used a proposed new computer mouse design, and while using the mouse, each student's wrist extension was recorded. Data consistent with summary values given in the paper "Comparative Study of Two Computer Mouse Designs" (Cornell Human Factors Laboratory Technical Report \(\mathrm{RP} 7992\) ) are given. Use these data to test the hypothesis that the mean wrist extension for people using this new mouse design is greater than 20 degrees. Are any assumptions required in order for it to be appropriate to generalize the results of your test to the population of Cornell students? To the population of all university students? \(\begin{array}{llllllllllll}27 & 28 & 24 & 26 & 27 & 25 & 25 & 24 & 24 & 24 & 25 & 28 \\ 22 & 25 & 24 & 28 & 27 & 26 & 31 & 25 & 28 & 27 & 27 & 25\end{array}\)
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