A rental agency claims that the average rent that small-business establishments pay in Suvacity is $800. A sample of 10
establishments shows an average rate of $863. The standard deviation of the sample is $20. At α = 0.05,is there enough evidence to reject the agencies
claim? Assume that the population is approximately normally distributed.
To determine whether there is enough evidence to reject the rental agency's claim, we can perform a hypothesis test.
Step 1: Formulate the null and alternative hypotheses:
The null hypothesis (H₀) is that the average rent in Suvacity is $800.
The alternative hypothesis (H₁) is that the average rent in Suvacity is not $800.
Step 2: Determine the significance level:
The significance level (α) is given as 0.05.
Step 3: Calculate the test statistic:
We can use the t-test statistic to compare the sample mean to the population mean when the population standard deviation is unknown. The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
Given:
x̄ (sample mean) = $863
μ (population mean) = $800 (claimed by the rental agency)
s (sample standard deviation) = $20
n (sample size) = 10
Substituting these values into the t-test formula, we get:
t = (863 - 800) / (20 / √10) = 2.57
Step 4: Determine the critical value:
Since we are conducting a two-tailed test (we are interested in deviations in both directions), we need to find the critical t-value at α = 0.05 / 2 = 0.025. With 9 degrees of freedom (n - 1), the critical t-value is approximately ±2.262.
Step 5: Compare the test statistic with the critical value:
If the absolute value of the t-test statistic is greater than the critical t-value, then we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, |2.57| > 2.262, so we reject the null hypothesis.
Step 6: Interpret the results:
Based on the sample data, there is enough evidence to reject the rental agency's claim that the average rent in Suvacity is $800.