A recent survey of 236 customers who frequently visit convenient stores in Kimberly found that the mean value of purchases was R 48.45, with the standard deviation of R18.58. Assume that the values are normally distributed. Construct a 90% confidence interval for the actual mean value of all purchases in convenient stores in Kimberly. Give the lower limit.
To construct a confidence interval for the mean value of all purchases in convenient stores in Kimberly, we can use the formula:
Confidence Interval = sample mean ± (critical value * (sample standard deviation / √sample size))
First, let's find the critical value for a 90% confidence interval. The critical value can be found using a z-table or a calculator. For a 90% confidence level, the critical value is 1.645.
Now we can calculate the confidence interval:
Confidence Interval = 48.45 ± (1.645 * (18.58 / √236))
Calculating the values inside the brackets:
(1.645 * (18.58 / √236)) ≈ 3.43
So the confidence interval is:
48.45 ± 3.43
The lower limit of the confidence interval is:
48.45 - 3.43 = 45.02
Therefore, the lower limit of the confidence interval for the actual mean value of all purchases in convenient stores in Kimberly is R 45.02.
To construct a confidence interval for the mean value of purchases in convenient stores in Kimberly, we can use the formula:
Confidence Interval = mean ± (Z * (standard deviation / √n))
Where:
- mean is the sample mean (R 48.45),
- Z is the Z-score corresponding to the desired confidence level (90% confidence level corresponds to a Z-score of 1.645),
- standard deviation is the sample standard deviation (R 18.58),
- n is the sample size (236).
Substituting the values into the formula, we calculate the confidence interval as follows:
Confidence Interval = 48.45 ± (1.645 * (18.58 / √236))
To calculate the standard error (standard deviation divided by the square root of the sample size), we have:
Standard Error = 18.58 / √236
Substituting the value:
Standard Error ≈ 1.21
Now, we can substitute the values into the confidence interval formula:
Confidence Interval = 48.45 ± (1.645 * 1.21)
Calculating the values:
Confidence Interval ≈ 48.45 ± 1.99
Finally, we can calculate the lower limit of the confidence interval by subtracting 1.99 from the mean:
Lower limit = 48.45 - 1.99 ≈ 46.46
Therefore, the lower limit of the 90% confidence interval for the actual mean value of purchases in convenient stores in Kimberly is approximately R 46.46.