91Ó°ÊÓ

Historically (from about 2001 to 2014 ), \(57 \%\) of Americans believed that global warming is caused by human activities. A March 2017 Gallup poll of a random sample of 1018 Americans found that 692 believed that global warming is caused by human activities. a. What percentage of the sample believed global warming was caused by human activities? b. Test the hypothesis that the proportion of Americans who believe global warming is caused by human activities has changed from the historical value of \(57 \%\). Use a significance level of \(0.01\). c. Choose the correct interpretation: i. In 2017 , the percentage of Americans who believe global warming is caused by human activities is not significantly different from \(57 \%\). ii. In 2017 , the percentage of Americans who believe global warming is caused by human activities has changed from the historical level of \(57 \%\).

Step by step solution

01

Calculate the sample percentage

In order to calculate the sample percentage, the number of Americans who believed in 2017 that global warming is caused by human activity should be divided by the total number of the 2017 sample and multiplied by 100. This gives the percentage of the 2017 sample that supports this belief. So the calculation is as follows: \((692 / 1018) * 100 = 68 \%\)

02

Set up the null and alternative hypotheses

To test whether the belief has changed from the historical value, there is a need to set up the null and alternative hypotheses. The null hypothesis (H0) represents the current status quo or the belief that there has been no change in the proportion. The alternative hypothesis (H1) suggests that there has been a change. Thus: H0: p = 0.57, H1: p ≠ 0.57.

03

Conduct the hypothesis test

First, the z-score is calculated. The z-score is equal to the difference between the sample proportion (p̂) and the hypothesized proportion (p0) divided by the standard error of the sampling distribution of p̂. The formulation is as follows: z = (p̂ - p0) / sqrt((p0 * (1 - p0)) / n), where n is the sample size. After the z score is found, the p-value must be calculated. The p-value is the probability that you would have found the current result, or a more significant result, by chance if the null hypothesis were true. Here, we must calculate a two-tailed p-value because the alternative hypothesis was two-sided. By using a Z-table or calculator, the p-value corresponding to the z-score can be found. Then this p-value is compared with the significance level to decide whether to reject the null hypothesis.

04

Interpret the result

Once the p-value is calculated, it needs to be compared with the significance level (α) to make a decision. If the p-value smaller than α (0.01), we reject H0, otherwise fail to reject H0. Then based on the decision made, choice i or ii should be chosen.

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Key Concepts

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

Proportion Hypothesis Test

A Proportion Hypothesis Test helps us determine if there's a significant difference between an observed sample proportion and a known or hypothesized population proportion.
In our present exercise, we're comparing a sample from 2017 against a historical belief from 2001 to 2014, which was that 57% of Americans thought global warming was driven by human activities.
To conduct this type of hypothesis test, we first need to find out what proportion of the sample shares this belief. That's where calculations come in, giving us clarity about whether our sample supports or refutes the established norm.

• The calculation involves dividing the number of individuals who hold a belief by the total sample size and then converting this into a percentage.

This exercise confirms the sample percentage from the data given before proceeding to test the hypothesis.

Significance Level

The significance level, often denoted by alpha (\(\alpha\)), quantifies the probability of rejecting the null hypothesis when it is actually true. In essence, it's the risk we are willing to take for making an incorrect decision.
For this hypothesis test, the significance level is set at 0.01, meaning we accept a 1% chance of mistakenly claiming a difference in beliefs about global warming when there is none.
This strict threshold implies stronger evidence is needed to reject the null hypothesis, protecting us against making premature conclusions based on random sample variations.

Null and Alternative Hypotheses

Setting up the null and alternative hypotheses is a fundamental step in hypothesis testing.
In the context of our exercise:

• The Null Hypothesis (H0) is the presumption of no change from the historical proportion, expressed as \( p = 0.57 \).
• The Alternative Hypothesis (H1) suggests a difference, articulating this as \( p eq 0.57 \) to allow for both an increase or decrease.

With these hypotheses established, we can proceed confidently into statistical testing. The null hypothesis serves as the baseline or default position, while the alternative seeks to prove deviation from it.

Z-Score Calculation

In the world of hypothesis testing, the Z-Score acts as a bridge to understanding where our sample stands in relation to the null hypothesis.
It quantifies the number of standard deviations our sample proportion \( \hat{p} \) lies from the hypothesized population proportion \( p_0 \).
To calculate this:

• Compute the standard error using the formula \( \sqrt{\frac{p_0 \times (1 - p_0)}{n}} \).
• Substitute the values into the Z-Score formula: \( Z = \frac{\hat{p} - p_0}{\text{Standard Error}} \).

This score allows us to derive the p-value and ultimately make our hypothesis decision.
The magnitude of the Z-Score lets us know how probable our sample proportion is under the assumption that the null hypothesis is true.