Question

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

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

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.

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.

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'.

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.

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.

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.

Key concepts

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 (p̂).