In 1930, the average size of a public restroom was 172 square feet. By 1990, due to disability laws, the average size had increased to 471 square feet. Suppose a design team believ... In 1930, the average size of a public restroom was 172 square feet. By 1990, due to disability laws, the average size had increased to 471 square feet. Suppose a design team believes that this standard has increased from the 1990 level. They randomly sample 23 public restrooms in a major midwestern city and obtained a mean footage of 498.78 with a standard deviation of 46.94. Test the hypothesis at α = 0.01 using the critical value rule. Read more

Steps to Solve

1. State the Hypotheses
   The null hypothesis ($H_0$) is that the average size of public restrooms is equal to 471 square feet. The alternative hypothesis ($H_a$) is that the average size has increased, i.e., $H_a: \mu > 471$.

2. Set the significance level and determine the critical value
   The significance level ($\alpha$) is 0.01. Since we are conducting a one-tailed test, we will find the critical value for a z-test. From the z-table, the critical z-value for $\alpha = 0.01$ is approximately $2.33$.

3. Calculate the test statistic
   We use the formula for the z-test statistic:

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

   where:

   • $\bar{x} = 498.78$ (sample mean)
   • $\mu_0 = 471$ (population mean under the null hypothesis)
   • $s = 46.94$ (sample standard deviation)
   • $n = 23$ (sample size)

   Substituting the values:

   $$ z = \frac{498.78 - 471}{\frac{46.94}{\sqrt{23}}} $$

4. Perform the calculations
   First, calculate the standard error:

   $$ SE = \frac{46.94}{\sqrt{23}} \approx 9.78 $$

   Now, calculate the z-test statistic:

   $$ z = \frac{498.78 - 471}{9.78} \approx 2.85 $$

5. Make a decision
   Compare the calculated z-value (2.85) with the critical value (2.33). Since $2.85 > 2.33$, we reject the null hypothesis.

6. Conclusion
   There is sufficient evidence at the $\alpha = 0.01$ significance level to support the claim that the average size of public restrooms has increased from the 1990 level.