91Ó°ÊÓ

In his book, "The Signal and the Noise," Nate Silver analyzed 733 predictions made by experts regarding political events. Of the 733 predictions, 338 were mostly true. (a) Determine the sample proportion of political predictions that were mostly true. (b) Suppose that we want to know whether the evidence suggests the political predictions were mostly true less thar half the time. State the null and alternative hypotheses. (c) Verify the normal model may be used to determine the \(P\) -value for this hypothesis test. (d) Draw a normal model with the area representing the \(P\) -val shaded for this hypothesis test. (e) Determine the \(P\) -value based on the model from part (d). (f) Interpret the \(P\) -value. (g) Based on the \(P\) -value, what does the sample evidence suggest? That is, what is the conclusion of the hypothesis test? Assume an \(\alpha=0.1\) level of significance.

Step by step solution

01

Part (a) - Determine the Sample Proportion

To find the sample proportion of political predictions that were mostly true, divide the number of mostly true predictions by the total number of predictions: \[ \hat{p} = \frac{338}{733} \] Simplify the fraction to obtain the sample proportion.

02

Part (b) - State the Hypotheses

To test if the evidence suggests that the political predictions were mostly true less than half the time, establish the null and alternative hypotheses. The null hypothesis (\(H_0\)) states that the population proportion of predictions that are mostly true is equal to 0.5. The alternative hypothesis (\(H_a\)) states that the population proportion of predictions that are mostly true is less than 0.5. Symbolically, \(H_0: p = 0.5\) and \(H_a: p < 0.5\).

03

Part (c) - Verify the Normal Model

Verify whether the normal model can be used by checking if the sample size is large enough. The conditions to check are: \(n \hat{p} \geq 10\) and \(n(1 - \hat{p}) \geq 10\). Calculate \(n \hat{p}\) and \(n(1 - \hat{p})\) to see if both values are greater than or equal to 10. If both conditions are satisfied, the normal approximation can be used.

04

Part (d) - Draw the Normal Model

Draw the normal distribution curve centered at the null hypothesis proportion (\(p = 0.5\)) with the standard error calculated as \( SE = \sqrt{\frac{p(1 - p)}{n}}\). Shade the area to the left of \(\hat{p}\). The shaded area represents the \(P\)-value for the hypothesis test.

05

Part (e) - Determine the P-value

Find the standard score (z-score) for the sample proportion using the formula: \[ z = \frac{\hat{p} - p}{SE} \] where \(SE = \sqrt{\frac{p(1-p)}{n}}\). Once the z-score is found, determine the corresponding \(P\)-value by looking up the z-score in the standard normal distribution table or using a calculator.

06

Part (f) - Interpret the P-value

The \(P\)-value represents the probability of observing a sample proportion as extreme as 0.461 if the null hypothesis is true. A small \(P\)-value indicates strong evidence against the null hypothesis.

07

Part (g) - Conclusion of the Hypothesis Test

Compare the \(P\)-value to the significance level (\(\alpha = 0.1\)). If the \(P\)-value is less than \(\alpha\), reject the null hypothesis. If the \(P\)-value is greater than \(\alpha\), do not reject the null hypothesis and suggest that the sample evidence is not strong enough to conclude that the proportion of mostly true predictions is less than 0.5.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Proportion

In hypothesis testing, the sample proportion is a critical concept that represents the proportion of successes in a sample. In our exercise, the sample proportion ( \( \hat{p} \) ) of political predictions that were mostly true is calculated by dividing the number of mostly true predictions by the total number of predictions. So, \( \hat{p} = \frac{338}{733} \approx 0.461 \) . This means about 46.1% of the predictions were mostly true. The sample proportion helps us to estimate what the true proportion might be in the entire population. This value is a primary input for further analysis in hypothesis testing.

Null Hypothesis

The null hypothesis ( \( H_0 \) ) is a statement we are trying to test. It typically represents the idea that there's no effect or no difference. In our scenario, the null hypothesis is: \( H_0: p = 0.5 \) , meaning the true proportion of mostly true predictions is 0.5 (or 50%). The null hypothesis acts as a starting point for statistical testing and is presumed true until evidence indicates otherwise. Rejecting or failing to reject the null hypothesis depends on how the sample data compares to this assumption.

Alternative Hypothesis

The alternative hypothesis ( \( H_a \) ) is the statement we want to find evidence for. It suggests that there is an effect or a difference. In our problem, the alternative hypothesis is: \( H_a: p < 0.5 \). This means we believe that the proportion of true political predictions is less than 0.5. The alternative hypothesis is directly tested through the data. If the data provides strong evidence against the null hypothesis, we accept the alternative hypothesis.

P-value

The \( P \)-value is a key concept in hypothesis testing, representing the probability that the observed data (or something more extreme) would occur if the null hypothesis were true. In this exercise, the \( P \)-value is found by calculating the z-score for the sample proportion and referencing the standard normal distribution. The z-score formula is: \( z = \frac{\hat{p} - p}{SE} \) , where \( SE \) is the standard error calculated as \( SE = \sqrt{\frac{p(1-p)}{n}} \). Once the z-score is found, the \( P \)-value is determined. If \( P \)-value is less than the significance level (\( \alpha = 0.1 \)), it indicates strong evidence against the null hypothesis.

Normal Distribution

In the context of hypothesis testing, the normal distribution plays a crucial role. It is a bell-shaped curve used to describe data that clusters around a mean. For our hypothesis test, we use the normal distribution because the sample size is large enough. We verify this by ensuring \( n\hat{p} \geq 10 \) and \( n(1 - \hat{p}) \geq 10 \). In this example, the conditions are satisfied since \( n\hat{p} = 338 \) and \( n(1 - \hat{p}) = 395 \). Therefore, we can use the normal approximation to determine the \( P \)-value , which helps in deciding whether to reject the null hypothesis.