Question

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

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

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 \( p > 0.5 \).

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: \( Z = (0.73 - 0.5) / \sqrt{(0.5*(1-0.5)/1000)} = 9.2 \).

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.

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.

Key concepts

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.