Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

The article referenced in the previous exercise also reported that 470 of 1000 randomly selected adult Americans thought that the quality of movies being produced was getting worse. a. Is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of .05 . b. Suppose that the sample size had been 100 instead of 1000 , and that 47 thought that the movie quality was getting worse (so that the sample proportion is still .47). Based on this sample of 100 , is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of .05 . c. Write a few sentences explaining why different conclusions were reached in the hypothesis tests of Parts (a) and (b).

Short Answer

Expert verified
To provide a legitimate answer, it's necessary to conduct the steps as outlined above which involve calculating test statistics, p-values and making decisions based on those values. The final explanation should address the influence of sample size on hypothesis testing outcomes.

Step by step solution

01

Set the Hypotheses

The null hypothesis (H0) is that p = 0.5, which stands for the idea that 50% (half) of all adult Americans believe that movie quality is getting worse. The alternative hypothesis (H1) is that p < 0.5, meaning that less than half of adult Americans believe that movie quality is getting worse.
02

Test statistics and p-value (Sample of 1000)

Use the formula for test statistic: z = (p̂ - p0) / sqrt ((p0 * (1 - p0)) / n), where p̂ (observed proportion) is 470 / 1000 = 0.47, p0 (expected proportion under H0) is 0.5, and n (sample size) is 1000. After calculation, find the corresponding p-value.
03

Make the Decision (Sample of 1000)

If p-value <= significance level (0.05), reject H0. Otherwise, fail to reject H0. The decision about the null hypothesis corresponds to the test result: if there is 'convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse'.
04

Test statistics and p-value (Sample of 100)

Now, repeat step 2 for the second case, where sample size decreased to 100 and the observed proportion is 47 / 100 = 0.47.
05

Make the Decision (Sample of 100)

Similar to step 3, make a decision based on the calculated p-value. Compare it to the significance level, if it's less or equal, reject H0, otherwise, fail to reject H0.
06

Explanation of Different Conclusions

Lastly, explain why different conclusions were reached based on the test results in step 3 and step 5. This often happens because sample size affects the results of a hypothesis test, influencing the statistical power to detect a difference from the assumed proportion 0.5.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Significance Level
When conducting hypothesis testing in statistics, the significance level is a critical concept that represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Setting the significance level acts as a threshold for how 'surprising' the data must be to warrant a rejection of the null hypothesis. A common significance level used is 0.05, meaning there is a 5% chance of committing a Type I error.

For example, in the exercise related to American opinions on movie quality, the significance level of 0.05 indicates that the researchers are willing to accept a 5% risk of concluding that less than half of the population believes movie quality is getting worse, when, in fact, half or more think so. This standard cutoff helps to provide consistency in statistical conclusions and enforces a balance between being too lenient and too strict in our decision-making process.
Null Hypothesis
The null hypothesis, denoted as H0, is a statement in hypothesis testing that there is no effect or no difference. It serves as the default assumption that a test seeks to challenge. In the given exercise, the null hypothesis is that the true population proportion (p) of adult Americans who believe that movie quality is getting worse is 0.5 or 50% (i.e., H0: p = 0.5).

It is important to conceptualize the null hypothesis as the skeptic's stance, which requires evidence to be overturned. Only when there is sufficient data suggesting that this hypothesis is unlikely, as per our significance level, can we reject it in favor of the alternative hypothesis that proposes a different scenario – in this case, that fewer than half of the population believe movie quality is worsening (H1: p < 0.5).
P-value
The p-value is a pivotal element in hypothesis testing as it measures the strength of the evidence against the null hypothesis. It gives the probability of observing test results at least as extreme as the actual observed results, assuming that the null hypothesis is correct. The smaller the p-value, the stronger the evidence against the null hypothesis.

Therefore, if the p-value is less than or equal to the chosen significance level (as in our example, 0.05), we reject the null hypothesis in favor of the alternative hypothesis. This implies that the observed data are highly unlikely to have occurred by random chance alone if the null hypothesis were true, giving us reason to believe in the alternative scenario proposed.
Sample Size Effect
The effect of sample size on hypothesis testing is a key consideration. A larger sample size reduces the standard error of the test statistic, which in turn can lead to more precise estimates and stronger evidence against the null hypothesis, provided there truly is an effect or difference to detect.

In the exercise, part (a) uses a sample of 1000, and part (b) uses a sample size of 100, maintaining the same observed proportion of 0.47. With a larger sample size, the test statistic usually has a clearer distinction from the null hypothesis, which can lead to a smaller p-value, making it easier to reject the null hypothesis. Conversely, with a smaller sample size, there's more uncertainty (higher standard error), and thus it may be harder to find convincing evidence to reject the null hypothesis, even with the same observed proportion. This concept is essential for understanding why different conclusions may be reached in situations like the parts (a) and (b) of the exercise, despite having the same observed statistic ().

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The report "Highest Paying Jobs for \(2009-10\) Bachelor's Degree Graduates" (National Association of Colleges and Employers, February 2010 ) states that the mean yearly salary offer for students graduating with a degree in accounting in 2010 is \(\$ 48,722\). Suppose that a random sample of 50 accounting graduates at a large university who received job offers resulted in a mean offer of \(\$ 49,850\) and a standard deviation of \(\$ 3300 .\) Do the sample data provide strong support for the claim that the mean salary offer for accounting graduates of this university is higher than the 2010 national average of \(\$ 48,722 ?\) Test the relevant hypotheses using \(\alpha=.05 .\)

The poll referenced in the previous exercise (“Military Draft Study," AP- lpsos, June 2005 ) also included the following question: "If the military draft were reinstated, would you favor or oppose drafting women as well as men?" Forty-three percent of the 1000 people responding said that they would favor drafting women if the draft were reinstated. Using a .05 significance level, carry out a test to determine if there is convincing evidence that fewer than half of adult Americans would favor the drafting of women.

The true average diameter of ball bearings of a certain type is supposed to be 0.5 inch. What conclusion is appropriate when testing \(H_{0}: \mu=0.5\) versus \(H_{a}: \mu \neq\) 0.5 inch each of the following situations: a. \(n=13, t=1.6, \alpha=.05\) b. \(n=13, t=-1.6, \alpha=.05\) c. \(n=25, t=-2.6, \alpha=.01\) d. \(\quad n=25, t=-3.6\)

"Most Like it Hot" is the title of a press release issued by the Pew Research Center (March 18, 2009 , www.pewsocialtrends.org). The press release states that "by an overwhelming margin, Americans want to live in a sunny place." This statement is based on data from a nationally representative sample of 2260 adult Americans. Of those surveyed, 1288 indicated that they would prefer to live in a hot climate rather than a cold climate. Do the sample data provide convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate? Use the nine-step hypothesis testing process with \(\alpha=.01\) to answer this question.

The amount of shaft wear after a fixed mileage was determined for each of seven randomly selected internal combustion engines, resulting in a mean of 0.0372 inch and a standard deviation of 0.0125 inch. a. Assuming that the distribution of shaft wear is normal, test at level .05 the hypotheses \(H_{0}: \mu=.035\) versus \(H_{\dot{a}}: \boldsymbol{\mu}>.035 .\) b. Using \(\sigma=0.0125, \alpha=.05,\) and Appendix Table 5, what is the approximate value of \(\beta,\) the probability of a Type II error, when \(\mu=.04\) ? c. What is the approximate power of the test when \(\mu=.04\) and \(\alpha=.05 ?\)

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free