Hypothesis Testing for One Sample Mean: Step-by-Step Guide

Hypothesis Testing for One Sample Mean

Overview of Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this example, we test whether the mean bounce height of new baseballs differs from a known value.

• Null Hypothesis (H0): The population mean is equal to the specified value.

• Alternative Hypothesis (H1): The population mean is different from the specified value.

• Significance Level (α): The probability threshold for rejecting H0, commonly set at 0.05.

Step-by-Step Procedure

Step 1: State the Hypotheses

Formulate the null and alternative hypotheses based on the claim.

• Original claim: μ ≠ 92.84

• H0: μ = 92.84

• H1: μ ≠ 92.84

Step 2: Calculate the Test Statistic

The test statistic measures how far the sample mean is from the hypothesized mean, in units of standard error. For a sample mean, use the t-test:

• Sample mean (x̄): 92.67

• Sample standard deviation (Sx): 1.79

• Sample size (n): 40

• Hypothesized mean (μ0): 92.84

The formula for the t-test statistic is:

$t = \frac{x\bar - \mu_0}{\frac{S_x}{\sqrt{n}}}$

For this example:

$t = \frac{92.67 - 92.84}{\frac{1.79}{\sqrt{40}}} \approx -0.601$

Step 3: Determine the P-value

The P-value is the probability of obtaining a result as extreme as the observed one, assuming the null hypothesis is true. Compare the P-value to the significance level (α).

• P-value: 0.5515

• α: 0.05

Step 4: State the Conclusion

If the P-value is greater than α, fail to reject the null hypothesis. If the P-value is less than or equal to α, reject the null hypothesis.

• Since 0.5515 > 0.05, we fail to reject H0.

• There is not enough evidence to support the claim that the new baseballs have bounce heights with a mean different from 92.84 inches.

Summary Table: Hypothesis Test Results

Parameter	Value
Sample Mean (x̄)	92.67
Sample Std. Dev. (Sx)	1.79
Sample Size (n)	40
Hypothesized Mean (μ0)	92.84
Test Statistic (t)	-0.601
P-value	0.5515
Significance Level (α)	0.05
Conclusion	Fail to reject H0

Example Application

This procedure is commonly used in quality control, scientific research, and business analytics to test claims about population means based on sample data.

Key Terms: Null hypothesis, alternative hypothesis, t-test, P-value, significance level.

Additional info: The t-test is appropriate when the population standard deviation is unknown and the sample size is moderate or large. The calculator screenshots illustrate the input and output for a one-sample t-test.