Question

5. True or False. (a) All other factors remaining the same, increasing the sample size, n, will decrease the width of a con dence interval. (b) We expect 95% of all 95% con dence intervals for the population mean to contain the sample mean. (c) The t-distribution is symmetric and centered at the population mean . (d) The t-distribution is very similar to the standard normal distribution regardless of its degrees of freedom. (e) All other factors remaining the same, a 90% con dence interval for a population mean is narrower than an 95% con dence interval for the same . 1

Answer 1

a) True

b) True

c) True

d) False

e) False

Explanation:

(a) All other factors remaining the same, increasing the sample size, n, will decrease the width of a confidence interval.

True. Confidence intervals are calculated by calculating margin of error (ME) around the mean using the formula

ME=
`(z*s)/(√(N) )` where

• z is the corresponding statistic (z-score or t-score)
• s is the standard deviation of the sample(or of the population if it is known)
• N is the sample size

As the formula suggests, all other factors remaining the same, if we increase N, ME decreases.

b) We expect 95% of all 95% confidence intervals for the population mean to contain the sample mean.

True. This is what 95% confidence level assumes.

(c) The t-distribution is symmetric and centered at the population mean

True.

(d) The t-distribution is very similar to the standard normal distribution regardless of its degrees of freedom.

False. As the degrees of freedom increases t-distribution resembles the standard normal distribution. For small sample sizes (<30), this is not true.

(e) All other factors remaining the same, a 90% confidence interval for a population mean is narrower than an 95% confidence interval for the same population

False. 90% confidence interval for a population mean is wider than an 95% confidence interval for the same population