91Ó°ÊÓ

Taking It Easy In a Gallup poll conducted December \(11-14,2003,455\) of 1011 randomly selected adults aged 18 and older said they had too little time for relaxing or doing nothing. (a) Verify that the requirements for constructing a confidence interval about \(\hat{p}\) are satisfied. (b) Construct a \(92 \%\) confidence interval for the proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing. Interpret this interval. (c) Construct a \(96 \%\) confidence interval for the proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing. Interpret this interval. (d) What is the effect of increasing the level of confidence on the width of the interval?

Step by step solution

01

Verify Requirements for Confidence Interval (a)

To verify the requirements for constructing a confidence interval about \(\hat{p}\), ensure the sample size is sufficient. The sample needs to satisfy the conditions of a binomial distribution: \(n\) is the number of trials (1011), and the sample proportion \(\hat{p} = \frac{455}{1011} \). We also must ensure \(np(1 - p) \geq 10\).

02

Calculate Sample Proportion

The sample proportion is calculated as \(\hat{p} = \frac{455}{1011} \approx 0.4505\).

03

Check Binomial Condition (a)

We need to check: \[ n\hat{p} = 1011 \times 0.4505 \approx 455 \] and \[ n(1 - \hat{p}) = 1011 \times 0.5495 \approx 556 \]. Both are greater than 10, so the requirements are satisfied.

04

Construct 92% Confidence Interval (b)

Use the formula for confidence interval: \[ \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] For a 92% confidence level, \ Z_{\alpha/2} \ approximate value is 1.75. Calculate: \[ 0.4505 \pm 1.75 \times \sqrt{\frac{0.4505(1-0.4505)}{1011}} \]. The interval is approximately \[ (0.422, 0.479) \]

05

Interpret the 92% Confidence Interval

About 92% of the time, the true proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing will fall within the interval (0.422, 0.479).

06

Construct 96% Confidence Interval (c)

For a 96% confidence level, \ Z_{\alpha/2} \ approximate value is 2.05. Calculate: \[ 0.4505 \pm 2.05 \times \sqrt{\frac{0.4505(1-0.4505)}{1011}} \]. The interval is approximately \[ (0.417, 0.484) \]

07

Interpret the 96% Confidence Interval

About 96% of the time, the true proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing will fall within the interval (0.417, 0.484).

08

Effect of Increasing Confidence Level (d)

Increasing the confidence level results in a wider confidence interval. This is because a higher confidence level requires a larger margin of error to ensure that the interval captures the true proportion.

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

The sample proportion, denoted as \(\hat{p}\), is a crucial concept when constructing confidence intervals. It represents the ratio of individuals or items in a sample that exhibit a particular characteristic. In our exercise, this was calculated as:
\[\hat{p} = \frac{455}{1011} \approx 0.4505\]
This means that approximately 45% of adults in the sample reported having too little time for relaxing.
Sample proportion helps us estimate the population proportion and is used in further statistical calculations.

Confidence Level

The confidence level tells us how sure we can be about the results of our statistical inference. It represents the percentage of all possible samples that can be expected to include the true population parameter.
A higher confidence level, such as 96%, means we are more certain that our interval contains the true population proportion, but this comes at a cost of a wider interval.
Conversely, a lower confidence level, like 92%, means the interval is narrower but we are less certain that it includes the true proportion.
You can think of the confidence level as a trade-off between certainty and precision.

Margin of Error

The margin of error shows how much the sample results are expected to vary from the true population value. When constructing a confidence interval, the margin of error is given by the formula:
\[ Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
This margin accounts for the variability in our sample. In our example, the margin of error for a 92% confidence interval was calculated using a Z-score value of approximately 1.75, resulting in a range of (0.422, 0.479). For a 96% confidence interval, a Z-score of 2.05 was used, resulting in a slightly broader range.
The margin of error grows with the confidence level because we need to cover more possible values of the true proportion, making us more confident about our results.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of trials. This distribution is applicable to our exercise because we are dealing with a binary outcome - either the adult has too little time for relaxing or they don't.
The binomial distribution assumptions are:

• Each trial has only two possible outcomes (success or failure).
• The probability of success, denoted as \(p\), is constant for each trial.
• The trials are independent.
• The number of trials, \(n\), is fixed.

For constructing confidence intervals, we verify the sample size meets the condition \( np(1-p) \geq 10 \). This ensures the sample distribution approximates a normal distribution, making our confidence interval calculations valid.

Z-Score

The Z-score is a statistical measurement that quantifies the number of standard deviations a data point is from the mean of a dataset. Z-scores are essential for finding confidence intervals around sample proportions.
The value of the Z-score varies depending on the confidence level. For example:

• For a 92% confidence level, the Z-score is approximately 1.75.
• For a 96% confidence level, the Z-score is approximately 2.05.

The general formula using the Z-score to calculate the confidence interval is:
\[ \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]This shows us how far from the sample proportion our intervals can stretch to confidently cover the true population proportion.
Understanding Z-scores helps in interpreting how the confidence intervals are formed and what the margins of error mean.