Question

# A simple random sample of size n is drawn from a population that is normally distributed....

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean,

x​,

is found to be

115​,

and the sample standard​ deviation, s, is found to be

10.

​(a) Construct

a

95​%

μ

if the sample​ size, n, is

22.

​(b) Construct

a

95​%

μ

if the sample​ size, n, is

12.

​(c) Construct

a

90​%

μ

if the sample​ size, n, is

22.

​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

​(a) Construct

a

95​%

μ

if the sample​ size, n, is

22.

Lower​ bound:

nothing​;

Upper​ bound: nothing

​(Use ascending order. Round to one decimal place as​ needed.)

​(b) Construct

a

95​%

μ

if the sample​ size, n, is

12.

Lower​ bound:

nothing​;

Upper​ bound: nothing

​(Use ascending order. Round to one decimal place as​ needed.)

How does

decreasing

the sample size affect the margin of​ error, E?

A.As the sample size

decreases​,

the margin of error stays the same.

B.As the sample size

decreases​,

the margin of error

decreases.

C.As the sample size

decreases​,

the margin of error

increases.

​(c) Construct

a

90​%

μ

if the sample​ size, n, is

22.

Lower​ bound:

nothing​;

Upper​ bound: nothing

​(Use ascending order. Round to one decimal place as​ needed.)

Compare the results to those obtained in part​ (a). How does

decreasing

the level of confidence affect the size of the margin of​ error, E?

A.As the level of confidence

decreases​,

the size of the interval

decreases.

B.As the level of confidence

decreases​,

the size of the interval stays the same.

C.As the level of confidence

decreases​,

the size of the interval

increases.

​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

A.

​Yes, the population needs to be normally distributed.

B.

​Yes, the population does not need to be normally distributed.

C.

​No, the population needs to be normally distributed.

D.

​No, the population does not need to be normally distributed.

a)

 sample mean 'x̄= 115 sample size    n= 22 std deviation s= 10 std error ='sx=s/√n=10/√22= 2.132
 for 95% CI; and 21 df, value of t= 2.080 from excel: t.inv(0.975,21) margin of error E=t*std error    = 4.434 lower bound=sample mean-E = 110.6 Upper bound=sample mean+E = 119.4 f

b)

 lower bound=sample mean-E = 108.6 Upper bound=sample mean+E = 121.4

C ).As the sample size decreases​, the margin of error increases.

c)

 lower bound=sample mean-E = 111.3 Upper bound=sample mean+E = 118.7

A.As the level of confidence decreases​,   the size of the interval decreases.

d) ​

A) Yes, the population needs to be normally distributed.

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