A large school district would like to estimate the mean score for all of its 5th grade students on a certain mathematics achievement test known to produce scores that are normally distributed in this population.

In a pilot study, n = 25 randomly selected 5th graders take the mathematics achievement test. The sample mean score is 63.4 points and the sample standard deviation is 7.2 points. Construct a 95% confidence interval for the mean score of all 5th graders in the district.
Check conditions
Since the o is unkown we will have to use the t-distribution. Our sample meets the requirements of being a normal population or n≥ 30 and is a random sample.
Critical values
t α/2 = ±2.064
Error
2.064 x 7.2/√25 = 1.44 x 2.064= 2.97216
Margin of error, E= 2.972
standard error= 1.44
Confidence interval
60.428 < μ < 66.372

b:The school board decides to conduct this study again for the next year's class of 5th graders. The school board would like to obtain a 99% confidence interval for the mean score of all 5th graders on the mathematics achievement test with a margin of error on this test of no more than 4 points. How large of a simple random sample should be taken?

How do you solve b?

bloop

To solve for b, we need to determine the sample size required to achieve the desired margin of error for the 99% confidence interval.

First, let's calculate the critical value for a 99% confidence interval. Since we are using the t-distribution and have a large enough sample size (unknown population standard deviation in this case), we can use the z-value instead of the t-value. The critical value for a 99% confidence interval is approximately 2.576.

Next, let's calculate the margin of error. The margin of error is the maximum distance allowed between the sample mean and the population mean within the confidence interval. In this case, we want the margin of error to be no more than 4 points.

Considering the margin of error formula:

Margin of error (E) = Critical value * Standard deviation / Square root of sample size

We can rearrange the formula to solve for the sample size:

Sample size = (Critical value * Standard deviation / Margin of error) ^ 2

Substituting the known values:

Sample size = (2.576 * 7.2 / 4) ^ 2

Calculating this equation gives you the approximate sample size needed for the next year's class of 5th graders to achieve a 99% confidence interval with a margin of error no more than 4 points.