Statistics students were asked to go home and fill a 1-cup measure with raisin bran, tap the cup lightly on the counter three times to settle the contents, if necessary add more raisin bran to bring the contents exactly up to the 1-cup line, spread the contents on a large plate, and count the 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
(a) Construct a 90 percent confidence interval for the mean number of raisins per cup. Show your work clearly. (b) Can you think of features of the sample or data-gathering method that might create problems? If so, how could they be improved? (c) Identify factors that might prevent Kellogg’s from achieving uniformity in the number of raisins per cup of raisin bran. (d) How might a quality
control system work to produce more uniform quantities of raisins, assuming that improvement is desired? Raisins
Calculate mean and standard deviation to find confidence interval (mean ± 1.645 SD). (SD value found in that same table.) For some of the other questions, be aware that, since raisins are heavier than the flakes, they are more likely to be found at the bottom of the cereal box.
I hope this helps.
(a) To construct a 90 percent confidence interval for the mean number of raisins per cup, we can follow these steps:
Step 1: Calculate the sample mean (x̄) and sample standard deviation (s) from the given data:
x̄ = (23 + 33 + 44 + 36 + 29 + 42 + 31 + 33 + 61 + 36 + 34 + 23 + 24) / 13 = 36.92 (rounded to two decimal places)
s = √[((23 - 36.92)^2 + (33 - 36.92)^2 + ... + (24 - 36.92)^2) / (13 - 1)] = 10.25 (rounded to two decimal places)
Step 2: Determine the critical value (z*) for a 90 percent confidence interval. We can use a z-table or a calculator to find this value. For a 90 percent confidence level, the critical value is approximately 1.645.
Step 3: Calculate the margin of error (E) using the formula:
E = z* * (s / √n), where n is the sample size.
In this case, n = 13.
E = 1.645 * (10.25 / √13) = 5.73 (rounded to two decimal places)
Step 4: Finally, construct the confidence interval using the formula:
Confidence interval = x̄ ± E
Confidence interval = 36.92 ± 5.73
Therefore, the 90 percent confidence interval for the mean number of raisins per cup is (31.19, 42.65).
(b) Possible features of the sample or data-gathering method that might create problems include:
- If the students did not measure accurately when filling the cup, it could introduce measurement errors.
- If the students did not tap the cup lightly on the counter three times consistently, it could affect settling of the contents.
- If the students did not add more raisin bran to exactly reach the 1-cup line, it could introduce variability in the measurements.
- If the students did not spread the contents evenly on the plate, it could affect the accuracy of counting the raisins.
To improve the data-gathering method, the following measures could be taken:
- Provide clear instructions to the students on how to accurately measure the raisin bran.
- Ensure consistency in tapping the cup on the counter to settle the contents.
- Use a measuring device with clear markings for the 1-cup line.
- Instruct the students to spread the contents evenly on the plate to facilitate accurate counting.
(c) Factors that might prevent Kellogg's from achieving uniformity in the number of raisins per cup of raisin bran could include:
- Natural variation in the size of raisins.
- Inconsistent distribution of raisins within the cereal box during the manufacturing process.
- Differences in the density of raisins within the cereal box.
- Variations in the amount of raisins added during the manufacturing process.
(d) A quality control system could work to produce more uniform quantities of raisins by implementing the following measures:
- Conducting regular inspections and quality checks during the manufacturing process to ensure consistent distribution of raisins within the cereal boxes.
- Implementing strict standards and guidelines for the amount of raisins to be added to each box of cereal.
- Using automated equipment or machinery to ensure accurate measurements of raisins during the manufacturing process.
- Implementing statistical process control techniques to monitor and minimize variability in the number of raisins per cup of cereal.
- Regularly collecting data on the number of raisins per cup and analyzing it to identify any patterns or trends that could be used to improve uniformity.