91Ó°ÊÓ

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).

Step by step solution

01

State the hypothesis

First, we need to state our null hypothesis \((H_0)\) and alternative hypothesis \((H_1)\). Since the problem asked if fewer than half believe that movie quality is getting worse, we know our null hypothesis is \(H_0: p = 0.5\), and our alternative hypothesis is \(H_1: p < 0.5\). \(p\) is the population proportion.

02

Step 2a: Conduct the test for a sample size of 1000

Now let's conduct our first test with a sample size of 1000 at a significance level of 0.05. Given the sample size \(n = 1000\) and the sample proportion \(\hat p = 470/1000 = 0.47\), we can now calculate our test statistic (Z score) and compare it with the critical Z value. Given that the standard deviation \(\sigma_{\hat p} = \sqrt{p(1-p)/n} = 0.0158\), our Z score \((\hat p - p)/\sigma_{\hat p} = (0.47 - 0.5)/0.0158 = -1.9\). Using a Z-table, the corresponding p-value associated with this Z score is around 0.0287 which is less than our significance level of 0.05. Thus we reject the null hypothesis.

03

Step 2b: Conduct the test for a sample size of 100

Repeat the same process for a sample size of 100. The standard deviation in this case is \(\sigma_{\hat p} = 0.05\), and our Z score \((\hat p - p)/\sigma_{\hat p} = (0.47 - 0.5)/0.05 = -0.6\). Looking this Z score up in a Z-table, we find that the corresponding p-value is approximately 0.2744. This is greater than our significance level of 0.05, hence we fail to reject the null hypothesis.

04

Explain the different conclusions

In the larger sample size, there was significant evidence to suggest that fewer than half of Americans believe that movie quality is getting worse. However, with a smaller sample size of 100, there was not enough evidence to conclude the same. This is because as our sample size decreases, our standard deviation increases, making our test statistic smaller and consequently leading to a larger p-value. The larger the p-value, the less evidence there is to reject the null hypothesis.

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Key Concepts

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

Null Hypothesis

In the context of hypothesis testing in statistics, the null hypothesis represents a default statement that there is no effect or no difference. It's a skeptical stance, presuming that any kind of observation is due to chance unless proven otherwise. For instance, if we are testing whether a new drug is more effective than the current treatment, the null hypothesis would state that the new drug has the same effect as the current one. In the exercise, the null hypothesis \(H_0: p = 0.5\) posits that exactly half of the adult Americans think the movie quality is not getting worse, hence indicating no difference from what's expected.

Alternative Hypothesis

The alternative hypothesis is the counter claim to the null hypothesis and it represents what the researcher is trying to prove. It is a statement that there is an effect or a difference. In the exercise, the alternative hypothesis \(H_1: p < 0.5\) suggests that fewer than half of adult Americans believe that movie quality is getting worse, which would be significant if proven, indicating a real perception of declining movie quality as opposed to chance. It's this hypothesis that researchers want to support with evidence from their data.

Significance Level

The significance level, often denoted by \(\alpha\), is the threshold for determining when to reject the null hypothesis. It's the probability of making the mistake of rejecting a true null hypothesis (a type I error). In simpler terms, it's how strong the evidence must be to conclude that the findings are not just due to random chance. A significance level of \(0.05\), used in the exercise, means there is a 5% risk of concluding that a difference exists when there is no actual difference.

Z-score

A Z-score is a statistical measurement that describes the relation of a single data point to the mean of a data set in units of standard deviation. In hypothesis testing, it's used to compare the observed results with what is expected under the null hypothesis. A higher absolute value of the Z-score indicates it's less likely the observed result occurred by chance. For the given exercise, it's calculated for each sample size and then used to determine the p-value that is compared to the significance level.

P-value

The p-value quantifies the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. A low p-value indicates that the observed data is unlikely under the null hypothesis, suggesting that the alternative hypothesis may be true. The p-value tells us how 'surprised' we should be by our data if the null hypothesis was true. In our exercise, for a sample size of 1000, the p-value was below the significance level, leading us to reject the null hypothesis.

Sample Size

The sample size is a critical component in any statistical analysis. It refers to the number of observations or replicates included in the study. Larger samples tend to give more accurate results because they better represent the population. They lead to smaller standard errors, increasing the likelihood that the test statistic will indicate a significant result if one truly exists. This concept is highlighted in the exercise where using a larger sample size of 1000 as opposed to 100 led to the rejection of the null hypothesis.

Population Proportion

The population proportion, in this case, refers to the actual proportion of adult Americans who believe the quality of movies is getting worse. The exercise deals with estimating this population proportion based on a sample. Since it's rarely feasible to measure an entire population, statisticians use samples to make inferences, which is why sample proportion \(\hat{p}\) is used as an estimate of the true population proportion \(p\).

Standard Deviation

The standard deviation is a measure that quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range. In hypothesis testing, the standard deviation of the sampling distribution, known as the standard error, determines how spread out the sample estimates are around the true population parameter. As shown in the exercise, a small sample size results in a larger standard deviation, which reduces the chances of observing a significant result.