The following data represents the percentages of family income allocated to
groceries for a random sample of 50 shoppers in Windhoek.
Percentages of family income Number of shoppers
10-<20 14
20-<30 6
30-<40 16
40-<50 3
50-<60 11
Construct a 99% confidence interval for the actual mean percentages of family
income allocated to groceries in Windhoek.
To construct a confidence interval for the mean percentages of family income allocated to groceries in Windhoek, we can use the following formula:
Confidence Interval = Sample Mean ± (Critical Value) × Standard Error
Step 1: Calculate the sample mean.
To calculate the sample mean, we need to find the average of the given percentages of family income allocated to groceries. We do this by multiplying each percentage by the corresponding number of shoppers and then summing them up. Finally, we divide this sum by the total number of shoppers.
Sample Mean = (10-<20 × 14) + (20-<30 × 6) + (30-<40 × 16) + (40-<50 × 3) + (50-<60 × 11) / Total Number of Shoppers
Step 2: Calculate the standard error.
The standard error measures the variability or dispersion of the sample mean. In this case, since we only have information about the number of shoppers in different income ranges, we would assume that the data follows a uniform distribution. Thus, we can calculate the standard error using the following formula:
Standard Error = (Range / (√12)) / Total Number of Shoppers
The range refers to any interval width, and since we have the intervals given, we can choose the upper limit of each interval. For this problem, we will choose 60 as the range.
Step 3: Calculate the critical value.
The critical value is determined by the confidence level and the degrees of freedom. Since we don't have the standard deviation of the population, we need to estimate it using the sample data. In this case, we have a small sample size, so we'll use the t-distribution.
The degrees of freedom for a sample size of 50 would be (Total Number of Shoppers - 1).
The critical value can be found using a t-distribution table or a statistical software.
Step 4: Plug the values into the formula.
Once we have the sample mean, standard error, and critical value, we can plug them into the formula to calculate the confidence interval.
Confidence Interval = Sample Mean ± (Critical Value) × Standard Error
For a 99% confidence interval, the critical value can be found using the t-distribution with a significance level of 0.01 (1 - confidence level).
Finally, you can substitute the values into the formula to obtain the confidence interval for the actual mean percentages of family income allocated to groceries in Windhoek.