Question

Information for Problems 1 – 9: The diameter of piston rings produced for automobile engines is known to be normally distributed with a population standard deviation,std, equal to 0.100 millimeters. The last ten piston rings produced by a particular manufacturer have the following diameters (in millimeters): 74.036, 74.432, 74.212, 74.071, 73.968, 74.231, 73.899, 74.035, 74.079 and 73.995

1.     If you were to calculate a confidence interval for the mean diameter of piston rings, would you use a Z-Interval or a T-Interval?

2.     If you were to calculate a two-sided 94% confidence interval on the mean diameter of piston rings, what would be the value of the |a_a/2| or |t_a/2| that you would use? Round to 3 decimal places.

3.     What is the two-sided 94% confidence interval on the mean diameter of piston rings given the data you have?

4.    Where does u fall in this interval? Where does x(line above) fall in this interval?

5.     What is the margin of error for this 94% confidence interval? In other words, what is the amount that you are adding and subtracting from x(line above)?

6.     If the manufacturer wants the error on this 94% confidence interval to be within 0.03 of the true mean, what sample size is needed?

7.     If the confidence level is decreased, would the width of the confidence interval increase or decrease? What variable in the formula is the cause of the increase or decrease?

8.     If you were to calculate a one-sided 94% confidence interval on the mean diameter of piston rings, what would be the value of the |z_a| or |t_a| that you would use?  

9.     Calculate the one-sided 94% Upper Bound on the mean diameter of piston rings.

Information for Problems 10 – 18: A civil engineer tests the compressive strength of 12 batches of concrete.   It is known that the compressive strength of concrete is normally distributed; however, the population standard deviation is unknown. The compressive strength is obtained from the following 12 readings (in psi): 2216, 2225, 2318, 2237, 2301, 2255, 2249, 2281, 2275, 2204, 2263, 2295

10. If you were to calculate a confidence interval for the mean compressive strength of the concrete, would you use a Z-Interval or a T-Interval?

11. If you were to calculate a two-sided 92% confidence interval for the mean compressive strength of the concrete, what would be the value of the |z_a/2| or |t_a/2| that you would use?  

12. What is the two-sided 92% confidence interval for the mean compressive strength of the concrete?

13. Where does u fall in this interval? Where does x(line above) fall in this interval?

14. What is the margin of error for this 92% confidence interval?

15. If the civil engineer wants the error on this 92% confidence interval to be within 15 psi of the true mean, what sample size is needed?

16. If you were to calculate a one-sided 92% confidence interval for the mean compressive strength of the concrete, what would be the value of the |z_a| or |t_a| that you would use?  

17. Calculate the one-sided 92% Lower Bound for the mean compressive strength of the concrete.

    

18. Consider the two-sided confidence interval again. If the sample size were increased, but the mean and standard deviation stayed the same, does the precision increase or decrease?

Information for Problems 19 – 24: An article in Knee Surgery, Sports Traumatology, Arthroscopy showed that of the 25 tears that were reported to be located between 3 mm and 6 mm from the meniscus, only 15 healed after surgery.

19. What is the sample proportion, p , of such tears that heal?

20. If you were to calculate a two-sided 89% confidence interval on the proportion of tears that heal, would you use |z_a/2| or |t_a/2| and what is the value?  

21. Calculate a two-sided 89% confidence interval on the proportion of such tears that will heal.   

22. If the researchers want the error on the 89% confidence interval to be within 0.10 of the true proportion of tears that heal after surgery, what sample size is needed? Use the estimated value, p , to do your calculations.

23. Repeat #22, but this time assume you do not have an estimated value for p.

24. Calculate a one-sided upper 89% confidence bound on the proportion of such tears that will heal.

Information for Problems 25 – 30: Suppose city police calculate the 95% confidence interval on the proportion of calls for drug overdoses that end in fatalities to be (0.59259, 0.92741).

25. What must be the value of p, the point estimate for the proportion of calls for drug overdoses that end in fatalities?

26. What must be the value of q?

27. What is the critical score that would be used to calculate the 95% confidence interval?

     

28. When can you use a T-score to calculate a confidence interval for a proportion?

29. If a sample size of 25 is used to calculate this confidence interval, how many fatalities must have occurred?

30. If the police want to be more precise in their estimate of the proportion of fatalities that result from drug overdoses, they should _____________ (increase/decrease) the level of confidence.

Answer 1

0.1557

1. If you were to calculate a confidence interval for the mean diameter of piston rings, would you use a Z-Interval or a T-Interval?
we calculate the Z interva since the population standard deviation,std, is given as 0.100 millimeters

2. If you were to calculate a two-sided 94% confidence interval on the mean diameter of piston rings, what would be the value of the |aa/2| or |ta/2| that you would use?
level of significance, alpha = 0.06
from standard normal table, two tailed z alpha/2 =1.881
since our test is two-tailed
reject Ho, if zo < -1.881 OR if zo > 1.881

3. What is the two-sided 94% confidence interval on the mean diameter of piston rings given the data you have?
given that,
standard deviation, σ =0.1
sample mean, x =74.0958
population size (n)=25
level of significance, α = 0.06
from standard normal table, two tailed z α/2 =1.881
since our test is two-tailed
value of z table is 1.881
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 74.0958 ± Z a/2 ( 0.1/ Sqrt ( 25) ) ]
= [ 74.0958 - 1.881 * (0.02) , 74.0958 + 1.881 * (0.02) ]
= [ 74.06,74.13 ]

4. Where does u fall in this interval? Where does x(line above) fall in this interval?
interpretations:
1. we are 94% sure that the interval [74.06 , 74.13 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 94% of these intervals will contains the true population mean

5. What is the margin of error for this 94% confidence interval? In other words, what is the amount that you are adding and subtracting from x(line above)?
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, α = 0.06
from standard normal table, two tailed z α/2 =1.881
since our test is two-tailed
value of z table is 1.881
margin of error = 1.881 * 0.02
= 0.04