Hypothesis Testing for a Population Mean

Hypothesis Testing for One Sample Mean

When to Use the z or t Distribution for Means

When testing a claim about a population mean, the choice between the normal (z) distribution and the t distribution depends on the sample size and whether the population standard deviation is known.

• Use the Normal z Distribution when:

  • The population distribution is normal or the sample size n > 30 (Central Limit Theorem applies).

  • The population standard deviation (σ) is known.

• Use the t Distribution when:

  • The population distribution is normal or the sample size n > 30.

  • The population standard deviation (σ) is unknown (use sample standard deviation s).

Note: For the t distribution, P-values are not always found in standard tables (such as Table A-3). In such cases, use the critical value method or statistical software/technology to determine P-values.

Test Statistics for Means

The test statistic measures how far the sample mean is from the hypothesized population mean, in terms of standard errors.

• z Test Statistic (when σ is known):

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

• t Test Statistic (when σ is unknown):

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

• Where:

  • \( \bar{x} \) = sample mean

  • \( \mu_0 \) = hypothesized population mean

  • \( \sigma \) = population standard deviation

  • \( s \) = sample standard deviation

  • \( n \) = sample size

Example 1: Testing a Claim About Cell Phone Radiation

Suppose we have measured radiation emissions (in W/kg) from a sample of cell phones. Assume the population is normally distributed. We want to test the claim that the mean radiation level is less than 1.00 W/kg at a 0.05 significance level.

• Step 1: State the Hypotheses

  • Null hypothesis: $H_0: \mu = 1.00$

  • Alternative hypothesis: $H_a: \mu < 1.00$

• Step 2: Identify the Test and Check Assumptions

  • Population is normal (given).

  • Use t-test if σ is unknown; use z-test if σ is known.

• Step 3: Calculate the Test Statistic

  • Use the appropriate formula for z or t (see above).

• Step 4: Find the P-value or Critical Value

  • For t-tests, use technology or the critical value method if tables are insufficient.

• Step 5: Make a Decision

  • If the test statistic falls in the critical region or the P-value is less than 0.05, reject the null hypothesis.

• Step 6: State the Conclusion

  • Interpret the result in the context of the claim (e.g., there is sufficient evidence to support the claim that the mean radiation is less than 1.00 W/kg).

Example 2

Another example would follow the same steps as above, using the provided summary statistics and the appropriate test (z or t) based on whether σ is known.

Summary Table: Choosing Between z and t Tests

Condition	Distribution to Use	Notes
Population normal or n > 30, σ known	z distribution	Use z-test formula
Population normal or n > 30, σ unknown	t distribution	Use t-test formula; degrees of freedom = n - 1

Additional info: In practice, most real-world problems use the t distribution because the population standard deviation is rarely known. The t distribution approaches the normal distribution as the sample size increases.