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Chapter 10: Problem 12

Haunted Houses In September \(1996,33 \%\) of adult Americans believed in haunted houses. In a Gallup Poll conducted June \(6-8,2005,370\) of 1002 adult Americans aged 18 or older believed in haunted houses. Is there sufficient evidence to conclude that the proportion of adult Americans who believe in haunted houses has increased at the \(\alpha=0.05\) level of significance?

Short Answer

Expert verified
Yes, there is sufficient evidence to conclude that the proportion of adult Americans who believe in haunted houses has increased since 1996.

Step by step solution

01

State the Hypotheses

We need to set up the null and alternative hypotheses. The null hypothesis (ull): The proportion of adults who believed in haunted houses in 2005 is the same as in 1996, \(H_0: p = 0.33\).The alternative hypothesis (ull): The proportion of adults who believed in haunted houses in 2005 is greater than in 1996, \(H_a: p > 0.33\).
02

Calculate the Test Statistic

We use the sample proportion to find the test statistic. The sample proportion is given by \( \frac{370}{1002} 鈮� 0.3693 \).Then, the test statistic is calculated using the formula \[ z = \frac{\bar{p} - p_0}{\frac{\bar{p}(1 - \bar{p})}{n}} \], where \( \bar{p} = 0.3693 \), \( p_0 = 0.33 \), and \( n = 1002 \).Plugging in the values, we get \[ z = \frac{0.3693 - 0.33}{\frac{0.33 (1-0.33)}{1002}} \].
03

Determine the Critical Value

Using the significance level \( ull = 0.05 \), we find the critical value from the standard normal distribution table. Since it's a one-tailed test, the critical value at \( ull = 0.05 \) is approximately 1.645.
04

Compare Test Statistic with Critical Value

Calculate the test statistic value. Substituting the values, \[ z 鈮� \frac{0.3693 - 0.33}{\frac{\bar{0.33}*0.67}{\bar{1002}}} 鈮� 3.05 \].Since 3.05 is greater than 1.645, we reject the null hypothesis.
05

State the Conclusion

As the test statistic exceeded the critical value, there is sufficient evidence to conclude that the proportion of adult Americans who believe in haunted houses has increased since 1996 at the 0.05 level of significance.

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Key Concepts

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

Null Hypothesis
When it comes to hypothesis testing, the **Null Hypothesis**鈥攄enoted as \( H_0 \)鈥攊s the default assumption that there is no effect or no difference. It represents the status quo or a statement of no change. In our exercise, the null hypothesis is that the proportion of adults believing in haunted houses in 2005 is the same as in 1996. Formally, this can be written as: \[ H_0: p = 0.33 \]. This means we start by assuming the belief percentage hasn't changed from the initial 33%.
Alternative Hypothesis
Conversely, the **Alternative Hypothesis**鈥攄enoted as \( H_a \)鈥攔epresents what we aim to support. It suggests that there is a significant effect or a difference. For our haunted houses example, the alternative hypothesis is that the proportion of adults believing in haunted houses in 2005 has increased compared to 1996. Formally, it can be framed as: \[ H_a: p > 0.33 \]. Here, the goal is to show that more than 33% of adult Americans now believe in haunted houses, indicating an increase in proportion.
Significance Level
The **Significance Level**鈥攄enoted as \( \alpha \)鈥攊s the threshold for deciding whether to accept or reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it's the risk we are willing to take for making a wrong decision. For our example, we set \( \alpha = 0.05 \). This means we have a 5% risk of concluding that the proportion of believers has increased when, in fact, it hasn't. Choosing a 0.05 significance level is common in many fields and indicates a moderate stance between being too liberal or conservative.
Critical Value
The **Critical Value** is a point on the scale of the test statistic beyond which we reject the null hypothesis. It depends on the significance level \( \alpha \) and the type of test (one-tailed or two-tailed). For a one-tailed test at \( \alpha = 0.05 \), the critical value from the standard normal distribution table is approximately 1.645. In our case, if the calculated test statistic exceeds 1.645, we'll reject the null hypothesis, indicating a statistically significant increase in the belief in haunted houses.
Test Statistic
The **Test Statistic** helps us determine whether there is sufficient evidence to reject the null hypothesis. It measures how far the sample data (observed proportion) deviates from the null hypothesis. Using the formula \[ z = \frac{\bar{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \], we plug in our values: \( \bar{p} = 0.3693 \), \( p_0 = 0.33 \), and \( n = 1002 \). After calculating, we get a test statistic of approximately 3.05. Since 3.05 is greater than the critical value of 1.645, we reject the null hypothesis, concluding a significant increase in belief in haunted houses at the 0.05 level of significance.

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