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Solutions to Problems on Confidence Intervals

Solutions to Problems on Confidence Intervals. BUAD820 Chapter 8. Confidence interval problem 1: Question 1. Sample mean. Known s. Sample size (n). Z-value for 99% CI. 99%CI runs from 0.9877 to 1.0023 gallons. Confidence interval problem 1: Question 1.

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Solutions to Problems on Confidence Intervals

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  1. Solutions to Problems on Confidence Intervals BUAD820 Chapter 8

  2. Confidence interval problem 1: Question 1 Sample mean Known s Sample size (n) Z-value for 99% CI 99%CI runs from 0.9877 to 1.0023 gallons

  3. Confidence interval problem 1: Question 1 • Why did we use the formula on page 287? • Because we know s (s known) • Because the sample size is large (n>30)

  4. Confidence interval problem 1: Question 2 • On the basis of your results, do you think that the manager has a right to complain to the manufacturer? Why? • NO! • That’s because the CI range contains the advertised value of 1 gallon • A CI range smaller than 1 gallon would cause concern • A CI range larger than 1 gallon would ..?

  5. Confidence interval problem 1: Question 3 • Does the population of paint per can have to be normally distributed (Hint: See Figure 7.5 on page 269)? • NO! • Per Figure 7.5 on page 269, we have a “large” sample (n>30) • Central Limit Theorem assures us that the distribution of sample means for large samples would be normal .. Thereby allowing us to use the CI formula

  6. Confidence interval problem 1: Question 4 • Why is an observed value of 0.98 gallon for an individual is not unlikely, although it is outside the confidence interval calculated above? • The can with 0.98 gallons represents just a single can • We are concerned with showing that the confidence interval contains the value 1

  7. Confidence interval problem 1: Question 5 • Which factors influence the size (width) of the confidence interval? • Level of confidence (90%, 95%, 99%) • Standard deviation (s) • Sample size (n)

  8. Confidence interval problem 1: Question 6 NOTE: We typically round sample size values up

  9. Confidence interval problem 2: Question 1 Sample standard deviation Sample size (n) Sample mean 95%CI = 184.66 to 205.94 t-value with 5% area in tails, and 17 degrees of freedom

  10. Confidence interval problem 2: Question 1 • Why did we use the formula on page 292? • Because we are dealing with: • Small sample (n=18), and, • Population standard deviation is unknown (s unknown; we used sample standard deviation, s, instead)

  11. Confidence interval problem 2: Question 2 • Do you think that the tires that do not meet the performance information provided on the sidewall? • NO • The value 200 is contained within the 95% confidence interval range

  12. Confidence interval problem 2: Question 3 Sample standard deviation Sample size (n) Sample mean 95%CI = 191.05 to 199.55 t-value with 5% area in tails, and 99 degrees of freedom

  13. Confidence interval problem 2: Question 3 • Why did the confidence interval change? • Standard error reduced as n went from 18 to 100 • t-value reduced because the sample size increased

  14. Confidence interval problem 3: Question 1 Sample proportion Standard error Z for 95% CI 95% CI runs from 12.23% to 22.77%

  15. Confidence interval problem 3: Question 2

  16. Confidence interval problem 3: Question 3 Tightening the CI requires larger n

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