The manufacture of the ColorSmart-5000 television set would like to claim 95 per

Question

The manufacture of the ColorSmart-5000 television set would like to claim 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 consumers who have owned a colorSmart-5000 television set for five years. Of these 400 consumers, 316 say their ColorSmart-5000 television sets did not need a repair, whereas 84 say their ColorSmart-5000 television sets did need at least one repair.

Problem 2: The manufacture of the ColorSmart-5000 television set would like to claim 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 consumers who have owned a colorSmart-5000 television set for five years. Of these 400 consumers, 316 say their ColorSmart-5000 television sets did not need a repair, whereas 84 say their ColorSmart-5000 television sets did need at least one repair. a) What is the sample proportion? Answer Canvas Quiz 6-4 (1 point) b) Find a 99 percent confident interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair. What is the critical value (Z-/2 or ton)? Answer Canvas Quiz 6-5. (1 point) What is the lower limit of the interval? Answer Canvas Quiz 6-6. (1 point) What is the upper limit of the interval? Answer Canvas Quiz 6-7. (1 point) c) Does this confidence interval provide strong evidence that the percentage of ColorSmart-5000 television sets that last at least five years without a single repair is less than the 95 percent claimed by the manufacturer? Circle your answer and explain. Use your answer to answer Canvas Quiz 6-8. (1 Point) Yes No Explain: d) Determine the sample size needed in order to be 99 percent confident that p, the sample proportion of ColorSmart-5000 television sets that last at least five years without a single repair, is within .03 of p, the true proportion of sets that last at least five years without a single repair. 1. Practical choice: Use your answer from part a) and answer Canvas Quiz 6-9. (1 point) 2. Conservative Choice: Use p = 0.5, and answer Canvas Quiz 6-10. (1 point)

Explanation / Answer

Answer:

a). P=0.79

b).

z=2.576

lower limit=0.7375

upper limit= 0.8425

CI = p pm z* sqrt {(p*(1-p)/n)}

Confidence Interval Estimate for the Proportion

Data

Sample Size

400

Number of Successes

316

Confidence Level

99%

Intermediate Calculations

Sample Proportion

0.79

Z Value

2.576

Standard Error of the Proportion

0.0204

Interval Half Width

0.0525

Confidence Interval

Interval Lower Limit

0.7375

Interval Upper Limit

0.8425

c).

Yes, confidence interval provides strong evidence because Both limits of 99% CI is less than 0.95.

d).

Sample size = (z2*p*(1-p))/d2

1). Practical approach = 1224

Sample Size Determination

Data

Estimate of True Proportion

0.79

Sampling Error

0.03

Confidence Level

99%

Intermediate Calculations

Z Value

2.576

Calculated Sample Size

1223.0326

Result

Sample Size Needed

1224.0000

c). conservative approach =1844

Sample Size Determination

Data

Estimate of True Proportion

0.5

Sampling Error

0.03

Confidence Level

99%

Intermediate Calculations

Z Value

2.576

Calculated Sample Size

1843.0268

Result

Sample Size Needed

1844.0000

Confidence Interval Estimate for the Proportion

Data

Sample Size

400

Number of Successes

316

Confidence Level

99%

Intermediate Calculations

Sample Proportion

0.79

Z Value

2.576

Standard Error of the Proportion

0.0204

Interval Half Width

0.0525

Confidence Interval

Interval Lower Limit

0.7375

Interval Upper Limit

0.8425

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