In Exercises 7–10, use the same population of {4, 5, 9} that was ...

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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 3m
- Introduction to Confidence Intervals
  15m
- Confidence Intervals for Population Mean
  48m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample1h 1m
- Steps in Hypothesis Testing
  1h 1m
10. Hypothesis Testing for Two Samples2h 8m
11. Correlation48m
- Scatterplots & Intro to Correlation
  26m
- Correlation Coefficient
  21m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

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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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 Variancea. Find the value of the population variance σ2.