An estate agent usually sells properties at the rate of 10 per week. During a recession, when mortgages are harder to get, he sells 63 properties over an eight-weekend period. Using a suitable approximation, test at the 5% level of significance whether or not there is evidence that the weekly sales has decreased..
To test whether there is evidence that the weekly sales have decreased during the recession, we can perform a hypothesis test using a suitable approximation. In this case, we will use a z-test.
Let's set up the hypotheses:
- Null Hypothesis (H0): The weekly sales rate during the recession is the same as the usual rate of 10 properties per week.
- Alternative Hypothesis (Ha): The weekly sales rate during the recession has decreased.
We will assume that the sample of 63 properties sold over an eight-weekend period during the recession is a representative sample.
To perform the z-test, we need to calculate the test statistic and compare it with the critical value.
Steps to follow:
Step 1: Calculate the sample mean and standard deviation.
- Sample mean (x̄) = total number of properties sold / number of weekends = 63 / 8 ≈ 7.875
- Sample standard deviation (s) = square root of [(number of properties sold - sample mean)^2 / (number of weekends - 1)] = square root of [(63 - 7.875)^2 / (8 - 1)] ≈ 9.028
Step 2: Set the significance level (α) and find the critical value.
- Significance level (α) = 0.05 (5% level of significance)
- Since the sample size is large (n = 8), we can use the standard normal distribution.
- The critical value for a two-tailed test at a 5% level of significance is approximately ±1.96
Step 3: Calculate the test statistic (z-score).
- Formula for the z-score: z = (x̄ - μ) / (s / sqrt(n))
- x̄ is the sample mean
- μ is the hypothesized population mean (10)
- s is the sample standard deviation
- n is the sample size
- Substitute the values into the formula: z = (7.875 - 10) / (9.028 / sqrt(8)) ≈ -1.829
Step 4: Compare the test statistic with the critical value and make a decision.
- Since the test statistic (-1.829) falls within the range of -1.96 to 1.96, we fail to reject the null hypothesis.
Interpretation:
Based on the data and the 5% level of significance, there is not enough evidence to conclude that the weekly sales rate has decreased during the recession.
Please note that this explanation assumes that the sample is representative and satisfies the assumptions of the z-test.