Statistics students were asked to go home and fill a 1-cup measure with raisin bran and to tap the cup gently on the counter 3 times to settle the contents. Then, if necessary, add more raisin bran to bring the contents exactly up to the 1-cup line. Once at the 1-cup line, the students would spread the contents on a large plate and count the number of raisins. For the 13 students who chose Kellogg’s brand, the reported results were:23 33 44 36 29 42 31 33 61 36 34 23 24

Question

Statistics students were asked to go home and fill a 1-cup measure with raisin bran and to tap the cup gently on the counter 3 times to settle the contents. Then, if necessary, add more raisin bran to bring the contents exactly up to the 1-cup line. Once at the 1-cup line, the students would spread the contents on a large plate and count the number of raisins. For the 13 students who chose Kellogg’s brand, the reported results were:23 33 44 36 29 42 31 33 61 36 34 23 24

Assuming the population is normal, construct a 90% confidence interval for the mean number of raisins per cup.

Answer 1

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.05) and its Z score.

90% = mean ± Z SEm

90% = mean ± 1.645 SEm

SEm = SD/√n

Find the mean = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

I'll let you do the calculations.

Answer 2

To construct a confidence interval for the mean number of raisins per cup, we can use the following steps:

Step 1: Calculate the sample mean (x̄) and sample standard deviation (s) from the given data.
- Sample Mean (x̄): Add up all the raisin counts and divide by the total number of students (13).
- Sample Standard Deviation (s): Calculate the standard deviation of the raisin counts.

Step 2: Calculate the standard error of the mean (SEM) using the formula: SEM = s / √n, where n is the sample size (13 in this case).

Step 3: Determine the critical value associated with the desired confidence level. For a 90% confidence level, the critical value is found using the t-distribution. Since the sample size is small (<30), we use a t-distribution rather than a z-distribution. The degrees of freedom for a sample size of 13 is 13 - 1 = 12.

Step 4: Calculate the margin of error (ME) using the formula: ME = critical value * SEM.

Step 5: Construct the confidence interval by subtracting and adding the margin of error to the sample mean: Confidence Interval = x̄ ± ME.

Let's go through the calculations:

Step 1:
Sample Mean (x̄) = (23 + 33 + 44 + 36 + 29 + 42 + 31 + 33 + 61 + 36 + 34 + 23 + 24) / 13 = 35.231

Sample Standard Deviation (s) = √[Σ(xi - x̄)² / (n - 1)] = √[Σ(xi²) - n(x̄)² / (n - 1)] = √[14495.1538 - (35.231)² / 12] = 11.783

Step 2:
SEM = s / √n = 11.783 / √13 = 3.267

Step 3:
Using a t-table or calculator, the critical value for a 90% confidence level and 12 degrees of freedom is approximately 1.782.

Step 4:
ME = critical value * SEM = 1.782 * 3.267 = 5.818

Step 5:
Confidence Interval = x̄ ± ME = 35.231 ± 5.818

Hence, the 90% confidence interval for the mean number of raisins per cup is (29.413, 41.050).