Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

Statistics

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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2:20 minutes

Problem 6.3.21

Textbook Question

In Exercise 11, would it be unusual for the population proportion to be 72.5%? Explain.

Verified step by step guidance

Step 1: Identify the context of the problem. The question is asking whether a population proportion of 72.5% is unusual. To determine this, we need to assess the concept of 'unusual' in statistics, which typically refers to values that are far from the mean (e.g., more than 2 standard deviations away).

Step 2: Recall the formula for the standard error of the proportion (SE). The formula is:

SE = sqrt((p (1 - p)) / n)

, where

p

is the population proportion and

n

is the sample size.

Step 3: Determine the z-score for the given proportion. The z-score formula is:

z = (p - p

₀) / SE, where

p

₀ is the hypothesized population proportion,

p

is the observed proportion, and

SE

is the standard error.

Step 4: Compare the z-score to the critical values for a standard normal distribution. Typically, a z-score beyond ±2 is considered unusual because it corresponds to the tails of the distribution (less than 5% of the data).

Step 5: Conclude whether the population proportion of 72.5% is unusual based on the z-score and the critical values. If the z-score is within ±2, it is not unusual; otherwise, it is unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Proportion

Population proportion refers to the fraction of a population that possesses a certain characteristic. It is a key parameter in statistics, often denoted as 'p', and is used to summarize categorical data. Understanding the population proportion helps in making inferences about the entire population based on sample data.

Statistical Significance

Statistical significance assesses whether the observed data deviates from what would be expected under a null hypothesis. A population proportion of 72.5% may be considered unusual if it falls outside the range of expected values based on sample data and confidence intervals. This concept is crucial for determining if a result is likely due to chance or represents a true effect.

Confidence Intervals

A confidence interval provides a range of values within which the true population parameter is expected to lie, with a certain level of confidence (e.g., 95%). If 72.5% falls outside this interval, it may indicate that this proportion is unusual or not representative of the population. Confidence intervals are essential for understanding the precision of estimates and making informed decisions based on sample data.

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Related Practice

Textbook Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.

b. Find the minimum sample size needed, using a prior study that found that 32% of U.S. adults have taken or planned to take a winter vacation in a recent year. (Source: Rasmussen Reports)

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Textbook Question

In Exercise 19, would it be unusual for the population proportion to be 38%? Explain.

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Textbook Question

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a. No preliminary estimate is available. Find the minimum sample size needed

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Textbook Question

b. Find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)

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Textbook Question

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Textbook Question

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Textbook Question

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.

(0.512, 0.596)

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