A rental agency claims that the average rent that small business establishments pay in Suva city is $800. A sample of 10 establishments shows an average rate of $863. The standard deviation of the sample is $20. At alpha = 0.05, is there enough evidence to reject the agencies claim? Assume that the population is approximately normally distributed.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Is it less than .05?
To determine if there is enough evidence to reject the rental agency's claim, we need to perform a hypothesis test.
Step 1: Formulate the null hypothesis (H0) and the alternative hypothesis (Ha).
- Null Hypothesis (H0): The average rent that small business establishments pay in Suva city is $800.
- Alternative Hypothesis (Ha): The average rent that small business establishments pay in Suva city is not $800.
Step 2: Select the appropriate test statistic. Since we have the sample mean, population mean, sample standard deviation, and the sample size is small (n < 30), we will use the t-test.
Step 3: Determine the significance level (alpha). In this case, alpha is given as 0.05.
Step 4: Calculate the test statistic. The formula for the t-test is:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is $863, the population mean is $800, the sample standard deviation is $20, and the sample size is 10.
t = (863 - 800) / (20 / sqrt(10))
Step 5: Determine the critical value(s). Since the alternative hypothesis is two-tailed (the average rent can be either higher or lower than $800), we will use a two-tailed t-distribution.
Using a t-distribution table or a calculator, with the degree of freedom (df) = sample size (n) - 1, which is 10 - 1 = 9, we find the critical values for a two-tailed test at alpha = 0.05 to be approximately +2.262 and -2.262.
Step 6: Compare the test statistic with the critical value(s).
If the absolute value of the test statistic is greater than the critical value(s), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, our test statistic is calculated to be t = 6.90 (rounded to two decimal places) using the formula above.
Since 6.90 > 2.262, we reject the null hypothesis.
Step 7: Interpret the results.
Based on our hypothesis test, there is enough evidence to reject the rental agency's claim that the average rent that small business establishments pay in Suva city is $800. This suggests that the average rent is significantly different from $800.