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In a simple random sample of 70 automobiles registered in

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 1E Chapter 5.2

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 1E

In a simple random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard.

a. What proportion of the automobiles in the sample had emission levels that exceed the standard?

b. Find a 95% confidence interval for the proportion of automobiles in the state whose emission levels exceed the standard.

c. Find a 98% confidence interval for the proportion of automobiles whose emission levels exceed the standard.

d. How many automobiles must be sampled to specify the proportion that exceed the standard to within ±0.10 with 95% confidence?

e. How many automobiles must be sampled to specify the proportion that exceed the standard to within ±0.10 with 98% confidence?

f. Someone claims that less than half of the automobiles in the state exceed the standard. With what level of confidence can this statement be made?

Step-by-Step Solution:

Step 1 of 5:

In a simple random sample of 70 automobiles that registered in a certain state,28 of them found to have a emission levels that exceeds a state standard.

We have to find

What proportion of automobiles in the sample had emission levels that exceed the standard.95 % CI for the proportion of automobiles in the state whose emission level exceeds the standard.98% CI for the proportion of automobiles in the state whose emission level exceeds the standard.How many automobiles must be sampled to specify the proportion that exceed the standard to within  with 95% confidence.How many automobiles must be sampled to specify the proportion that exceed the standard to within  with 98% confidence.At what level of confidence the claim that less than  half of the automobiles in the state exceed the standard.

Step 2 of 5:

The proportion of automobiles in the samples had emission levels that exceed the standard.

It is given that,

Total number of automobiles registered in a certain state = 70.

Number of automobiles that exceed the standard             = 28.

 

The proportion of automobiles in the sample whose emission level exceed the standard

                                                                 =   

                                                              = 0.4

Therefore proportion of automobiles in the sample had emission levels that exceeds the standard is 0.4 (or 40%).

  (b) The 95% confidence interval for the proportion of automobiles whose emission level exceed the standard.

 Let X be the number of success in an independent bernoulli trials with success probability P, so that X~B(n,P).

Lets define = n+4.

And           =  

Then a level 100(1-confidence interval for P is

                                                

Let X denote the number of automobiles that exceed the standard,and n is the total number of  automobiles.

                 X   =  28

                      n    =  70

                   70+4

                           =  74

And

                     

                          =  0.4054

Since here the Z-score is =

We can find from the standard normal table,

So the 95% confidence interval for the proportion of automobiles is

                               ( 0.4054 )

                         (  0.2935   , 0.517 )

Therefore the 95% confidence interval for the proportion of automobiles is (0.2935, 0.517).

Step 3 of 4

Chapter 5.2, Problem 1E is Solved
Step 4 of 4

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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