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 if there is evidence of a decrease in weekly sales during the recession, we can use a hypothesis test.
Step 1: Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The weekly sales rate during the recession is the same as the usual rate, 10 properties per week.
- Alternative hypothesis (Ha): The weekly sales rate during the recession is less than 10 properties per week.
Step 2: Determine the test statistic:
Since we are dealing with proportions (sales per week), we can use the z-test as an appropriate approximation. The test statistic is calculated as:
z = (x - μ) / (σ / sqrt(n))
where:
- x is the sample mean (sales during the recession over eight weeks)
- μ is the population mean (usual weekly sales rate, 10 in this case)
- σ is the population standard deviation (unknown in this case)
- n is the sample size (eight weeks)
Step 3: Set the level of significance:
The significance level (alpha) is given as 0.05 or 5%. This means we want to be 95% confident in our results.
Step 4: Determine the critical value:
Since this is a one-tailed test (we are testing if sales have decreased), we will find the critical value for a one-tailed test at the 5% significance level. The critical value for a 5% level of significance is -1.645.
Step 5: Calculate the test statistic:
Calculate the sample mean (x), which is the average sales during the recession over eight weeks.
x = 63 / 8 = 7.875
Since we don't know the population standard deviation (σ), we need to estimate it using the sample standard deviation (s). In this case, we don't have access to the individual sample data, so we can use the approximation σ ≈ s.
The sample standard deviation (s) can be calculated using the sample mean (x) and the sample size (n):
s = sqrt((∑(x - x̄)^2) / (n - 1))
where ∑(x - x̄)^2 is the sum of squared differences between each data point and the sample mean (squared deviation).
Since we only have the information given in the question, we can't calculate the sample standard deviation (s) directly. However, for large sample sizes, it is reasonable to assume that the sample standard deviation is close to the population standard deviation.
Step 6: Calculate the test statistic (z):
z = (x - μ) / (σ / sqrt(n))
Since σ ≈ s, we can replace σ with s in the formula:
z = (x - μ) / (s / sqrt(n))
Substituting the values we have:
z = (7.875 - 10) / (s / sqrt(8))
Step 7: Make a decision:
Compare the calculated test statistic with the critical value: -1.645.
If the calculated test statistic (z) is less than the critical value (-1.645), we reject the null hypothesis (H0) and conclude that there is evidence that the weekly sales rate has decreased during the recession. If the calculated test statistic is greater than or equal to the critical value, we fail to reject the null hypothesis and conclude that there is no evidence of a decrease in weekly sales.
Step 8: Perform calculations:
Since we don't have the sample data, we can't calculate the sample standard deviation and the test statistic. However, if you have the necessary data, you can substitute the values into the formulas and perform the calculations.
In conclusion, to test whether there is evidence of a decrease in weekly sales during the recession, you need to calculate the test statistic using the formulas provided and compare it with the critical value. If the test statistic is less than the critical value, you reject the null hypothesis and conclude that there is evidence of a decrease in sales rate.