Question

A simple random sample of size n is drawn. The sample​ mean, x overbar x​, is found to be 17.8​, and the sample standard​ deviation, s, is found to be 4.9.

​(a) Construct a 95​% confidence interval about mu μ if the sample​ size, n, is 34.

Lower​ bound=

Upper​ bound=

​(Use ascending order. Round to two decimal places as​ needed.)

​(b) Construct a 95​% confidence interval about mu μ if the sample​ size, n, is 61.

Lower​ bound=​

Upper​ bound=

​(Use ascending order. Round to two decimal places as​ needed.)

How does increasing the sample size affect the margin of​ error, E?

A. The margin of error increases.

B. The margin of error does not change.

C. The margin of error decreases.

​(c) Construct a 99 99​% confidence interval about mu μ if the sample​ size, n, is 34.

Lower​ bound=

Upper​ bound=

​(Use ascending order. Round to two decimal places as​ needed.)

Compare the results to those obtained in part​ (a). How does increasing the level of confidence affect the size of the margin of​ error, E?

A. The margin of error decreases.

B. The margin of error does not change.

C. The margin of error increases.

​(d) If the sample size is 18​, what conditions must be satisfied to compute the confidence​ interval?

A. The sample size must be large and the sample should not have any outliers.

B. The sample must come from a population that is normally distributed and the sample size must be large.

C. The sample data must come from a population that is normally distributed with no outliers.

Answer 1

Answer :

Given data is :

Mean = = 17.8

Standard deviation = s = 4.9

a)Given sample size n = 34

Confidence level is at 95%

Z value at 95% is 1.96

CI =

=

=

=

=

=

=

CI = (16.15 , 19.45)

Lower bound = 16.15

Upper bound = 19.45

b)1)Given sample size n = 61

Confidence level is at 95%

Z value at 95% is 1.96

CI =

=

=

=

=

=

=

CI = (16.57 , 19.03)

Lower bound = 16.57

Upper bound = 19.03

b)2)Based on these,

If we increase the sample size then the margin of error will decrease.

Option C is correct.

Because,Margin of error ME=

in part a) n = 34 then ME = 1.6464

in part b) n = 61 then ME = 1.229

c)1)Given sample size n = 34

Confidence level is at 99.99%

Z value at 99.99% is 3.891

CI =

=

=

=

=

=

=

CI = (14.53 , 21.07)

Lower bound = 14.53

Upper bound = 21.07

C)2)Based on these values,

If confidence interval increases then the margin of error increases

Option C is correct.

d)Given sample size is 18.

i.e It is less than 30

The sample data must come from a population that is normally distributed with no outliers.

Option C is correct.