18. A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 50, and the sample standard deviation, s, is found to be 8. (a) Construct a 98% confidence interval for m if the sample size, n, is 20. (b) Construct a 98% confidence interval for m if the sample size, n, is 15. How does decreasing the sample size affect the margin of error, E? (c) Construct a 95% confidence interval for m if the sample size, n, is 20. Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, E? (d) Could we have computed the confidence intervals in parts (a)–(c) if the population had not been normally distributed? Why?

d. No. because the confidence interval coefficient is gotten from the table of normal distribution of Z value. therefore the confidence interval is only used for normal distributed population

Step-by-step explanation:

given

The sample mean, x, = 50,

the sample standard deviation, s, = 8.

a. Construct a 98% confidence interval for m if the sample size, n, is 20

for a 98% confidence interval of an inﬁnite population, thus for a sample of size N and mean x, drawn from an inﬁnite population having a standard deviation of σ, the mean value of the population is estimated to by:

x± 2.33σ /√N

for a conﬁdence level of 98%, Zc = 2.33 gotten from the table of confidence interval.

This indicates that the mean value of the population lies between

50- ( 2.33*8 /√20) and 50+ ( 2.33*8 /√20)

with 98% conﬁdence in this prediction.

=50 ± 4.67

=45.33 or 54.67

(b) Construct a 98% confidence interval for m if the sample size, n, is 15.

using the same method as above

x± 2.33σ /√N

This indicates that the mean value of the population lies between

50- ( 2.33*8 /√15) and 50+ ( 2.33*8 /√15)

=50 ± 4.81

=45.19 or 54.81

decreasing the sample size increase the margin of error

(c) Construct a 95% confidence interval for m if the sample size, n, is 20. Compare the results to those obtained in part

95% confidence interval =1.96 (gotten from table of confidence coefficient)

using x±1.96 σ /√N

This indicates that the mean value of the population lies between

50- (1.96 *8 /√20) and 50+ (1.96 *8 /√20)

=50 ± 3.51

=46.49 or 53.51

c. decreasing the confidence level reduced the margin of error

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

No. because the confidence interval coefficient is gotten from the table of normal distribution of Z value. therefore the confidence interval is only used for normal distributed population