Hypothesis Testing with One Sample: Z-Test for the Mean (σ Known)

Hypothesis Testing with One Sample

Introduction to Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha), and using sample evidence to determine which hypothesis is more plausible.

• Null Hypothesis (H0): A statement that there is no effect or no difference; it is assumed true until evidence suggests otherwise.

• Alternative Hypothesis (Ha): A statement that contradicts the null hypothesis; it represents the claim to be tested.

• Level of Significance (α): The probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05, 0.01, or 0.10.

Hypothesis Testing for the Mean (σ Known)

When the population standard deviation (σ) is known, the z-test is used to test hypotheses about the population mean (μ). This test is appropriate when the sample is random and either the population is normally distributed or the sample size is large (n ≥ 30).

• Test Statistic: The standardized test statistic for the mean is given by:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• Where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the hypothesized population mean, \( \sigma \) is the population standard deviation, and n is the sample size.

Using P-values to Make a Decision

The P-value is the probability, assuming the null hypothesis is true, of obtaining a result equal to or more extreme than what was actually observed. The decision rule is as follows:

• If P-value ≤ α, reject the null hypothesis (H0).

• If P-value > α, fail to reject the null hypothesis.

Example: If the P-value is 0.0237 and α = 0.05, since 0.0237 < 0.05, reject H0. If α = 0.01, since 0.0237 > 0.01, fail to reject H0.

Finding the P-value for a Hypothesis Test

The method for finding the P-value depends on the type of test:

• Left-tailed test: P-value is the area to the left of the test statistic.

• Right-tailed test: P-value is the area to the right of the test statistic.

• Two-tailed test: P-value is twice the area in the tail beyond the absolute value of the test statistic.

Example (Left-tailed): For z = -2.24, the area to the left is 0.0125. If α = 0.01, fail to reject H0 because 0.0125 > 0.01.

Example (Two-tailed): For z = 2.14, area to the right is 0.0162, so P-value = 2 × 0.0162 = 0.0324. If α = 0.05, reject H0 because 0.0324 < 0.05.

z-Test for a Mean (μ)

The z-test for a mean is used when the population standard deviation is known. The test statistic is calculated as shown above. The test is valid if the sample is random and either the population is normally distributed or the sample size is large.

Examples of Hypothesis Testing Using P-values

• Pit Stop Example: A pit crew claims its mean pit stop time is less than 13 seconds. Sample mean = 12.9, n = 32, σ = 0.19. Calculated z and P-value. If P-value < α, reject H0 and support the claim.

• Home Temperature Example: Testing if the mean indoor temperature at night is different from a known value. Use two-tailed test, calculate z and P-value, and compare with α to make a decision.

Rejection Regions and Critical Values

The rejection region (or critical region) is the range of values for which the null hypothesis is rejected. The critical value separates the rejection region from the non-rejection region.

• For a left-tailed test, the rejection region is to the left of the critical value.

• For a right-tailed test, the rejection region is to the right of the critical value.

• For a two-tailed test, the rejection regions are in both tails beyond the critical values.

Finding Critical Values:

• Specify α and the type of test (left, right, two-tailed).

• Find the z-score(s) corresponding to the area(s) in the tail(s).

Example: For a left-tailed test with α = 0.01, the critical value is z = -2.33. For a two-tailed test with α = 0.05, the critical values are z = -1.96 and z = 1.96.

Using Rejection Regions for a z-Test for the Mean (σ Known)

To use the rejection region approach:

1. Calculate the standardized test statistic (z).

2. Compare z to the critical value(s):

   • If z is in the rejection region, reject H0.

   • If z is not in the rejection region, fail to reject H0.

Example: Testing if the mean salary of mechanical engineers is less than $105,200. Sample mean = $103,000, n = 20, σ = $9,500. Calculate z, compare to critical value, and make a decision based on the rejection region.

Summary Table: Decision Rules for Hypothesis Testing (z-Test, σ Known)

Test Type	Rejection Region	Critical Value(s)	Decision Rule
Left-tailed	z < -zα	-zα	Reject H0 if z < -zα
Right-tailed	z > zα	zα	Reject H0 if z > zα
Two-tailed	|z| > zα/2	±zα/2	Reject H0 if |z| > zα/2

Key Terms and Concepts

• P-value: Probability of observing a test statistic as extreme as, or more extreme than, the observed value under H0.

• Critical Value: The value that separates the rejection region from the non-rejection region.

• Rejection Region: The set of values for the test statistic that leads to rejection of H0.

• Level of Significance (α): The threshold probability for rejecting H0.

Additional info: The image included is the cover of the textbook 'Elementary Statistics' by Ron Larson, which is directly relevant as it is the source of the material summarized here.