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Math  /  Data & Statistics

Question1. As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective. a) What population is under consideration in the data set? b) What parameter is being estimated? c) What is the point estimate for the parameter? d) What is the name of the statistic we use to measure the uncertainty of the point estimate? e) Compute the value from part (d) for this context. f) The historical rate of defects is 10%10 \%. Should the engineer be surprised by the observed rate of defects during the current week?

Studdy Solution

STEP 1

1. The total number of chips sampled is 212.
2. The number of defective chips found is 27.
3. The historical defect rate is 10%10\%.

STEP 2

1. Identify the population under consideration.
2. Identify the parameter being estimated.
3. Calculate the point estimate for the parameter.
4. Identify the statistic used to measure the uncertainty of the point estimate.
5. Compute the value of the statistic from part (d).
6. Compare the observed defect rate with the historical rate to determine if the engineer should be surprised.

STEP 3

a) The population under consideration is all computer chips produced by the factory during the week of production.

STEP 4

b) The parameter being estimated is the proportion of defective chips in the entire population of chips produced during the week.

STEP 5

c) The point estimate for the parameter is the sample proportion of defective chips. Calculate the sample proportion:
p^=Number of defective chipsTotal number of chips sampled=27212\hat{p} = \frac{\text{Number of defective chips}}{\text{Total number of chips sampled}} = \frac{27}{212}
Calculate the value:
p^≈0.1274\hat{p} \approx 0.1274

STEP 6

d) The statistic used to measure the uncertainty of the point estimate is the standard error of the sample proportion.

STEP 7

e) Compute the standard error of the sample proportion:
SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
where n=212 n = 212 and p^≈0.1274 \hat{p} \approx 0.1274 .
Calculate the standard error:
SE≈0.1274×(1−0.1274)212≈0.0223SE \approx \sqrt{\frac{0.1274 \times (1 - 0.1274)}{212}} \approx 0.0223

STEP 8

f) Compare the observed defect rate with the historical rate:
The observed defect rate is p^≈12.74% \hat{p} \approx 12.74\% , and the historical defect rate is 10%10\%.
To determine if the engineer should be surprised, consider if the observed rate is significantly different from the historical rate. Calculate the z-score:
z=p^−0.10SE≈0.1274−0.100.0223≈1.23z = \frac{\hat{p} - 0.10}{SE} \approx \frac{0.1274 - 0.10}{0.0223} \approx 1.23
A z-score of 1.23 is not typically considered statistically significant at common significance levels (e.g., α=0.05 \alpha = 0.05).
Therefore, the engineer should not be surprised by the observed rate of defects.

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