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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 6 - Confidence IntervalsProblema 6.2.15
Capitolo 6, Problema 6.2.15

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

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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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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.
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Critical Values: t-Distribution

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.
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Calculating the Mean

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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Critical Values: t-Distribution
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