1 . Suppose a 95% confidence interval for the mean is (10, 12) when you know the value of the population standard deviation (sigma). If you HAD NOT KNOWN the value of sigma, would your confidence interval have been wider, narrower, or the same?

 

2 . Suppose your hypotheses in a hypothesis test for the mean are Ho: u = 8 vs Ho: u > 8. Suppose your test statistic using Z is 2.13 and you knew the value of the population standard deviation. If you HAD NOT KNOWN the value of the population standard deviation, would your test statistic using t have been greater, smaller, or the same? 

 

3 . (T/F) There are situations where if you had known the population standard deviation you could have rejected Ho, but since you did not know it, you ended up failing to reject Ho.

 

Answer:

(1) ->

In confidence interval, if the critical value is higher, the confidence interval will be wider and vice versa.

We use Standard normal distribution critical value when population standard deviation is known and t distribution critical value when population standard deviation is unknown.

Since the t_critical ?≥? Z_critical.

Hence if we HAD NOT KNOWN the value of the population standard deviation, the confidence interval will be wider.

(2) ->

In hypothesis test, test statistics is NOT dependent on critical value and is dependent on population or sample standard deviation.

If the value of sample standard deviation is greater than population standard deviation then value of test statistics will be smaller and vice versa. Also test statistic remains same if both sample and population standard deviation are equal.

In the question, it is told that population standard deviation is UNKNOWN and it doesn't mention any value of standard deviation. So, we are assuming that the sample standard deviation is equal to population standard deviation which wll results in the same test statistcs.

Hence Test statistics using t will be same.

(3) ->

There are situations where if you had known the population standard deviation you could have rejected Ho, but since you did not know it, you ended up failing to reject Ho. True.

Because For the same value "x",

P(Z > x) ?≤? P(t > x).

Which may results in the p-value is less than level of significance resulting in rejecting the null hypothesis.