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

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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1:40 minutes

Problem 6.2.15

Textbook Question

In Exercises 15 and 16, find the t-value for the given values of xbar, μ, s and n.
xbar = 70.3, μ = 64.8, s = 7.1, n = 16

Verified step by step guidance

Step 1: Recall the formula for the t-value in a one-sample t-test:

t = \frac{x̄ - μ}{s / \sqrt{n}}

, where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Step 2: Substitute the given values into the formula: x̄ = 70.3, μ = 64.8, s = 7.1, and n = 16. The formula becomes:

t = \frac{70.3 - 64.8}{7.1 / \sqrt{16}}

Step 3: Simplify the denominator by calculating the standard error of the mean:

\text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{7.1}{\sqrt{16}}

. Compute the square root of 16 and divide 7.1 by the result.

Step 4: Subtract the population mean (μ) from the sample mean (x̄):

70.3 - 64.8

. This gives the numerator of the t-value formula.

Step 5: Divide the result from Step 4 (numerator) by the result from Step 3 (denominator) to calculate the t-value:

t = \frac{\text{Numerator}}{\text{Denominator}}

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Key Concepts

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

t-value

The t-value is a statistic used in hypothesis testing to determine if there is a significant difference between the sample mean and the population mean. It is calculated by taking the difference between the sample mean (x̄) and the population mean (μ), and dividing it by the standard error of the mean. The t-value helps assess how far the sample mean is from the population mean in terms of standard deviations.

Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) quantifies how much the sample mean (x̄) is expected to vary from the true population mean (μ). It is calculated by dividing the sample standard deviation (s) by the square root of the sample size (n). A smaller SEM indicates that the sample mean is a more accurate estimate of the population mean, which is crucial for calculating the t-value.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in a statistical calculation. In the context of a t-test, the degrees of freedom are typically calculated as n - 1, where n is the sample size. This concept is important because it affects the shape of the t-distribution used to determine critical values and p-values in hypothesis testing.

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(14.7, 22.1)

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