Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

Statistics

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

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2:41 minutes

Problem 6.3.7a

Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Variance

a. Find the value of the population variance σ².

Verified step by step guidance

Step 1: Recall the formula for population variance (σ²), which is given by:

σ^{2} = \frac{\sum (x - μ)^{2}}{N}, where μ is the population mean, x represents each data point, and N is the population size.

Step 2: Calculate the population mean (μ) using the formula:

μ = \frac{\sum}{x} . For the population {4, 5, 9}, sum the values (4 + 5 + 9) and divide by the population size (N = 3).

Step 3: Subtract the population mean (μ) from each data point in the population to find the deviations: (x - μ). Then, square each deviation to get (x - μ)².

Step 4: Sum all the squared deviations obtained in Step 3. This gives the numerator of the variance formula:

\sum (x - μ)^{2}

Step 5: Divide the sum of squared deviations by the population size (N = 3) to calculate the population variance (σ²).

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Key Concepts

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

Population Variance

Population variance is a measure of how much the values in a population differ from the population mean. It is calculated by taking the average of the squared differences between each data point and the mean. For the population {4, 5, 9}, the variance quantifies the spread of these values, providing insight into the variability within the entire population.

Sampling Distribution

The sampling distribution of a statistic, such as the sample variance, describes the distribution of that statistic across all possible samples of a given size from a population. When samples are taken with replacement, each sample can yield different values, and the sampling distribution helps to understand the variability and expected behavior of the sample variance as more samples are drawn.

Random Sampling with Replacement

Random sampling with replacement means that each time a sample is drawn from the population, the selected element is returned to the population before the next draw. This method ensures that each element has an equal chance of being selected in every draw, which is crucial for maintaining the independence of samples and for accurately estimating population parameters like variance.

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Multiple Choice

A previous study found that $80 percent sign$ of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that, from this same random sample of $100$ people, that between $10$ and $11$ people prefer Coca Cola.

Textbook Question

Fatal Car Crashes There are about 15,000 car crashes each day in the United States, and the proportion of car crashes that are fatal is 0.00559 (based on data from the National Highway Traffic Safety Administration). Assume that each day, 1000 car crashes are randomly selected and the proportion of fatal car crashes is recorded.

a. What do you know about the mean of the sample proportions?

Textbook Question

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.

a. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)?

Textbook Question

b. What value do the sample means target? That is, what is the mean of all such sample means?

Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

c. Find the mean of the sampling distribution of the sample variance.

Textbook Question

Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where and Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

Textbook Question

MCAT The Medical College Admissions Test (MCAT) is used to help screen applicants to medical schools. Like many such tests, the MCAT uses multiple-choice questions with each question having five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of “a.”

b. Find the mean of the sampling distribution of the sample proportion.

Textbook Question

Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.

a. After identifying the 25 different possible samples, find the proportion of peas with yellow pods in each of them, then construct a table to des

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