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Approval Rating for Congress In a Gallup poll \(^{51}\) conducted in December 2015 , a random sample of \(n=824\) American adults were asked "Do you approve or disapprove of the way Congress is handling its job?" The proportion who said they approve is \(\hat{p}=0.13,\) and a \(95 \%\) confidence interval for Congressional job approval is 0.107 to 0.153 . If we use a 5\% significance level, what is the conclusion if we are: (a) Testing to see if there is evidence that the job approval rating is different than \(14 \%\). (This happens to be the average sample approval rating from the six months prior to this poll.) (b) Testing to see if there is evidence that the job approval rating is different than \(9 \%\). (This happens to be the lowest sample Congressional approval rating Gallup ever recorded through the time of the poll.)

Step by step solution

01

State the Null and Alternative Hypotheses

(a) Here we are testing to see if there is evidence that the job approval rating is different than 14%. Thus, the null hypothesis (\(H_0\)) is that the approval rating (\(p\)) equals 14%, \(H_0: p = 0.14\), and the alternative hypothesis (\(H_a\)) is that the approval rating (\(p\)) is different from 14%, \(H_a: p \neq 0.14\).\n\n(b) Similarly, here we are testing if the approval rating is different than 9%. So, the null hypothesis (\(H_0\)) is that the approval rating equals 9%, \(H_0: p = 0.09\), and the alternative hypothesis is that the approval rating is different than 9%, \(H_a: p \neq 0.09\).

02

Interpret the Confidence Interval

A 95% confidence interval for approval is from 0.107 to 0.153. This tells us that based on this sample, we are 95% confident that the true approval rating lies within this interval.

03

Conclusions of the Tests

(a) Because 14% is within our confidence interval (0.107 to 0.153), we do not reject the null hypothesis and conclude that there is not enough evidence to say that the approval rating is different than 14%.\n\n(b) Because 9% is not within our confidence interval (0.107 to 0.153), we reject the null hypothesis and conclude that there is enough evidence to say that the approval rating is different than 9%.

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Key Concepts

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

Understanding Confidence Intervals

A confidence interval gives us a range in which we expect the true value of a parameter to lie. In the context of the Gallup poll, the interval from 0.107 to 0.153 tells us about Congress's approval rating.
We are 95% confident that the actual approval rating is within this range.

Confidence intervals are calculated using sample data and are used to make inferences about a larger population. The width of the interval depends on several factors:

• Sample size: Larger samples tend to give more precise (narrower) intervals.
• Variability: More variability in the data can widen the interval.
• Confidence level: Increasing the confidence level (e.g., from 95% to 99%) will also widen the interval.

This particular confidence interval helps us assess how likely specific approval percentages are, such as 14% or 9%.

The Role of the Null Hypothesis

The null hypothesis ( H_0 ) is a statement used in hypothesis testing that presumes no effect or no difference. It acts as a starting point for statistical significance testing.
In our exercise, two null hypotheses are tested:

• For 14%: The null hypothesis states that H_0: p = 0.14 , meaning there is no difference from this percentage.
• For 9%: Here, H_0: p = 0.09 , implying the rating is not different from 9%.

By comparing the null hypothesis to the confidence interval, we determine if evidence supports rejecting or accepting it. If the proposed percentage falls within the confidence interval, we typically don't reject the null hypothesis, as seen with 14%.
Conversely, if it falls outside, like the 9% case, we reject the null hypothesis, signaling a significant difference.

Deciphering the Significance Level

The significance level, often denoted by \( \alpha \), is a critical threshold in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error).
In the Gallup poll, a significance level of 5% (or 0.05) was used. This means we're willing to accept a 5% chance of incorrectly rejecting the null hypothesis.

The significance level plays a key role in hypothesis testing:

• A higher significance level increases the chance of detecting an effect when there is one, but also the risk of a false positive.
• A lower significance level decreases this risk but requires stronger evidence to reject the null hypothesis.

It's crucial to choose a significance level that balances sensitivity and specificity based on research goals and potential consequences of errors. In our exercise, this 5% threshold helped decide whether the observed approval rates were significantly different from 14% and 9%.