1. A random sample of size 49 is taken from a large population, measuring the time it takes to complete a driver’s license examination. The sample mean was found to be 47 minutes, and the sample standard deviation 5.89 minutes. Construct a 95% confidence interval around the sample mean
95% interval = mean ± 1.96 SEm
SEm = SD/√(n-1), but you can just use n
I hope this helps.
45.35 to 48.65
(z test interval) =45.35 to 48.65
To construct a confidence interval around the sample mean, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard error)
1. First, let's find the critical value. Since we want to construct a 95% confidence interval, we need to find the z-score at a confidence level of 95%. The critical value can be found using a standard normal distribution table or a calculator. For a 95% confidence level, the critical value (z-score) is approximately 1.96.
2. Next, we calculate the standard error. The standard error represents the variability of the sample mean. It can be calculated using the formula: standard deviation / sqrt(sample size).
In this case, the sample standard deviation is 5.89 minutes, and the sample size is 49.
Standard error = 5.89 / √(49)
3. Now, substitute the values into the confidence interval formula:
Confidence Interval = 47 ± (1.96 * (5.89 / √49))
4. Calculate the standard error using the equation:
√49 = 7
5. Plug in the values:
Confidence Interval = 47 ± (1.96 * (5.89 / 7))
6. Calculate the values within the parentheses:
5.89 / 7 = 0.8414
7. Calculate the confidence interval:
Lower Limit = 47 - (1.96 * 0.8414)
Upper Limit = 47 + (1.96 * 0.8414)
Lower Limit = 47 - 1.65
Upper Limit = 47 + 1.65
Lower Limit = 45.35 minutes
Upper Limit = 48.65 minutes
Therefore, the 95% confidence interval around the sample mean is (45.35, 48.65) minutes.