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

1. Step 1: Recall the formula for the t-value in a one-sample t-test: $t\; =\; \backslash frac\{x̄\; -\; \mu \}\{s\; /\; \backslash sqrt\{n\}\}$, where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
2. Step 2: Substitute the given values into the formula: x̄ = 70.3, μ = 64.8, s = 7.1, and n = 16. The formula becomes: $t\; =\; \backslash frac\{70.3\; -\; 64.8\}\{7.1\; /\; \backslash sqrt\{16\}\}$.
3. Step 3: Simplify the denominator by calculating the standard error of the mean: $\backslash text\{Standard\; Error\}\; =\; \backslash frac\{s\}\{\backslash sqrt\{n\}\}\; =\; \backslash frac\{7.1\}\{\backslash sqrt\{16\}\}$. Compute the square root of 16 and divide 7.1 by the result.
4. 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.
5. Step 5: Divide the result from Step 4 (numerator) by the result from Step 3 (denominator) to calculate the t-value: $t\; =\; \backslash frac\{\backslash text\{Numerator\}\}\{\backslash text\{Denominator\}\}$.

Key Concepts

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.