Question

The article referenced in the previous exercise also reported that 470 of 1,000 randomly selected adult Americans thought that the quality of movies being produced is getting worse. a. Is there convincing evidence that less than half of adult Americans believe that movie quality is getting worse? Use a significance level of 0.05 . b. Suppose that the sample size had been 100 instead of \(1,000,\) and that 47 thought that movie quality was getting worse (so that the sample proportion is still 0.47 ). Based on this sample of 100 , is there convincing evidence that less than half of adult Americans believe that movie quality is getting worse? Use a significance level of 0.05 . c. Write a few sentences explaining why different conclusions were reached in the hypothesis tests of Parts (a) and (b).

Short answer

a) We would reject the null hypothesis if our calculated z-score is less than -1.645. b) Similarly, we reject the null hypothesis if the z-score calculated (with n=100) is less than -1.645. c) Different conclusions are reached because of the different sample sizes. The larger sample size gives a more precise estimate of the population proportion, therefore providing stronger evidence.

Step-by-step solution

Hypothesis formulation

The null hypothesis (H0) is that half or more of Americans think movie quality is getting worse. Hence, H0: p >= 0.5. The alternative hypothesis (H1) is that less than half of Americans believe the movie quality is declining. Therefore H1: p < 0.5.

Calculate the z-score for a)

Since we know that the sample size is 1,000 and 470 out of 1,000 think the quality is decreasing, then the sample proportion (p̂) is 0.47. We can use the z-score formula and replace known values. The z score formula is: z = (p̂ - p0) / \(\sqrt{((p0(1 - p0)) / n)}\), where \(p0\) is the population proportion under the null hypothesis, \(p0 = 0.5\), \(p̂\) is the sample proportion, hence \(p̂ = 0.47\) and n is the sample size, hence n = 1,000.

Find the critical z-score for a)

As the significance level (alpha) is 0.05 and it's a one-tail test, hence the critical z-score is the z-score relating the top 5% of the distribution. Using a Z table, the critical z-score for alpha 0.05 is -1.645.

Comparison and decision for a)

Compare calculated z-score with the critical z-score. If the calculated z-score is less than the critical z-score we reject the null hypothesis.

Decide for a

The decision should be based on the comparison in step 4

Calculate the z-score for b)

In this part, the sample size is 100 but the sample proportion is still 0.47. Replace the values into the z score formula.

Find the critical z-score for b)

For the significance level of 0.05 and one-tail test, the critical z-score remains the same as -1.645.

Comparison and decision for b)

Just like in steps 4 & 5, Compare and make a decision based on the z-scores

Conclusions for c

Different conclusions are tied directly to different sample sizes in a) and b), even with the same proportions. The larger sample gives a better estimate of the population proportion, if the sample is representative. Consequently, the same evidence might be strong or weak depending on the sample size.