Given a random sample of 144 with a mean of 100 and a standard deviation of 60 is taken
from a population of 1000. Find the confidence interval using a level of significance of
10% and 5%.
To find the confidence interval, we need to use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
where:
- Sample Mean: the mean of the sample
- Critical Value: determined by the level of significance and the sample size
- Standard Error: calculated using the sample standard deviation and sample size
Let's calculate the confidence intervals for a level of significance of 10% and 5%:
1. Confidence Interval with a level of significance of 10%:
Step 1: Calculate the critical value (z-score) for a 10% level of significance:
We need to find the z-score that corresponds to a 10% level of significance from the standard normal distribution (z-distribution). This can be done using a z-table or a statistical software. For a 10% level of significance, the z-score is approximately 1.645.
Step 2: Calculate the standard error:
Standard Error (SE) = Standard Deviation / √Sample Size
SE = 60 / √144 = 60 / 12 = 5
Step 3: Calculate the confidence interval:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 100 ± (1.645 * 5)
Confidence Interval = 100 ± 8.225
Confidence Interval = (91.775, 108.225)
Therefore, the confidence interval with a level of significance of 10% is (91.775, 108.225).
2. Confidence Interval with a level of significance of 5%:
Step 1: Calculate the critical value (z-score) for a 5% level of significance
For a 5% level of significance, the z-score is approximately 1.96.
Step 2: Calculate the standard error (same as before):
Standard Error (SE) = 5
Step 3: Calculate the confidence interval:
Confidence Interval = 100 ± (1.96 * 5)
Confidence Interval = 100 ± 9.8
Confidence Interval = (90.2, 109.8)
Therefore, the confidence interval with a level of significance of 5% is (90.2, 109.8).
Keep in mind that the confidence interval represents the range of values within which we are confident that the true population mean lies. In this case, we are 90% confident that the population mean lies within the interval (91.775, 108.225) and we are 95% confident that the population mean lies within the interval (90.2, 109.8).