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
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To construct a 90 percent confidence interval for the mean number of raisins per cup, we can use the t-distribution since the sample size is small (n = 13).
a) To calculate the confidence interval, we need the sample mean, the sample standard deviation, and the t-score.
1. Sample mean (x̄):
The sample mean is calculated by summing up all the observations and dividing by the sample size: x̄ = (23 + 33 + 44 + 36 + 29 + 42 + 31 + 33 + 61 + 36 + 34 + 23 + 24) / 13 = approx. 35.85.
2. Sample standard deviation (s):
Next, we need to calculate the sample standard deviation. The formula for the sample standard deviation is:
s = √(Σ(xi - x̄)^2 / (n - 1)),
where xi represents each individual observation.
(xi - x̄)^2 = (23-35.85)^2 + (33-35.85)^2 + (44-35.85)^2 + (36-35.85)^2 + (29-35.85)^2 + (42-35.85)^2 + (31-35.85)^2 + (33-35.85)^2 + (61-35.85)^2 + (36-35.85)^2 + (34-35.85)^2 + (23-35.85)^2 + (24-35.85)^2.
Σ(xi - x̄)^2 = the sum of all (xi - x̄)^2.
Then, divide Σ(xi - x̄)^2 by (n - 1) and take the square root to get the sample standard deviation.
3. Degrees of freedom (df):
The degrees of freedom is given by (n-1), where n is the sample size. In this case, df = 13 - 1 = 12.
4. T-score for 90 percent confidence interval:
To find the T-score for a 90 percent confidence interval with 12 degrees of freedom, we can use a t-distribution table or a statistical software. The T-score for a 90 percent confidence interval and 12 degrees of freedom is approximately 1.782.
Now, we can calculate the margin of error:
Margin of error = t-score * (s/√n).
Finally, we can construct the confidence interval:
Confidence interval = sample mean ± margin of error.
b) Possible problems with the sample or data-gathering method could include:
- Variation in the tapping method used by different students, leading to inconsistent settling of the raisin bran contents.
- Differences in cup sizes used by the students, resulting in varying amounts of raisin bran being measured.
- Human errors in counting the number of raisins accurately.
To improve these issues, the instructions for the students could be standardized to ensure consistency in tapping, cup sizes, and counting methods. Additionally, the experiment could be repeated multiple times to reduce random errors.
c) Factors that might prevent Kellogg's from achieving uniformity in the number of raisins per cup could include:
- Natural variation in the distribution of raisins within the cereal.
- Inconsistent manufacturing or processing methods that result in differing amounts of raisins in each batch.
- Quality control issues during packaging and distribution that could lead to inconsistent amounts of raisins per cup.
d) To produce more uniform quantities of raisins, a quality control system could:
- Implement stricter manufacturing and processing standards to ensure more accurate and consistent distribution of raisins within each batch.
- Conduct regular inspections and measurements during packaging to identify and reject packages that do not meet the desired uniformity standards.
- Use automated systems to measure and dispense precise quantities of raisin bran.
- Perform regular checks on the calibration of measuring instruments to ensure accuracy.