A simple random sample of 25 has been collected from a normally distributed population for which it is know that o=17.0. The sample mean has been calculated as 342.0, and the sample standard deviation is s=14.9. Construct and interpret the 95% and 99% confidence intervals for the population mean.
CI95 = mean + or - 1.96(sd divided by √n)
...where + or - 1.96 represents the 95% confidence interval using a z-table, sd = standard deviation, √ = square root, and n = sample size.
For the 99% confidence interval, you will have to substitute the value from a z-table that corresponds to 99%.
Put the values you are given from the problem into the formula and go from there to determine your confidence intervals.
I hope this will help get you started.
To construct a confidence interval for the population mean, you can use the formula:
Confidence interval = sample mean +/- (critical value * standard error)
Where:
- Sample mean is the mean of your sample (342.0)
- Critical value is the value obtained from the t-distribution table, which depends on the desired confidence level and the degrees of freedom
- Standard error is calculated as the sample standard deviation divided by the square root of the sample size
Let's start with the 95% confidence interval:
1. Degrees of freedom:
Since you have a sample size of 25, the degrees of freedom for a t-distribution is 25-1 = 24.
2. Critical value:
To find the critical value for a 95% confidence level with 24 degrees of freedom, you can consult the t-distribution table or use a statistical software. For this example, the critical value is approximately 2.064.
3. Standard error:
Standard error = sample standard deviation (s) / square root of the sample size
Standard error = 14.9 / sqrt(25)
Standard error = 14.9 / 5
Standard error = 2.98
Now let's construct the 95% confidence interval:
Confidence interval = 342.0 +/- (2.064 * 2.98)
Confidence interval = 342.0 +/- 6.14
Confidence interval = (335.86, 348.14)
Interpretation of the 95% confidence interval:
We can say with 95% confidence that the true population mean falls within the range of 335.86 and 348.14.
To construct the 99% confidence interval, you can follow the same steps, but with a different critical value.
4. Critical value:
For a 99% confidence level with 24 degrees of freedom, the critical value is approximately 2.797.
Now let's construct the 99% confidence interval:
Confidence interval = 342.0 +/- (2.797 * 2.98)
Confidence interval = 342.0 +/- 8.35
Confidence interval = (333.65, 350.35)
Interpretation of the 99% confidence interval:
We can say with 99% confidence that the true population mean falls within the range of 333.65 and 350.35.
Remember, a higher confidence level results in a wider confidence interval because it includes a larger range of values.