Question

A 90% confidence interval is constructed for the population mean. If a 95% confidence interval had been constructed instead (everything else remaining the same), the width of the interval would have been ________ and the probability of making an error would have been _________.

Answer 1

Answer:

If a 95% confidence interval had been constructed instead (everything else remaining the same), the width of the interval would have been wider and the probability of making an error would have been smaller.

Step-by-step explanation:

The width of a confidence interval is:

$W=2\times CV\times \frac{\sigma}{\sqrt{n}}$

The width depends on three things,

• Sample size (n)

• Sample standard deviation (σ)

• Critical value (CV)

The critical value of the test statistic is dependent upon the confidence level.

The higher the confidence level larger is the critical value and vice-versa.

And the critical value is directly proportional to the width of the confidence interval.

So, bigger the confidence level wider is the interval.

For instance, refer the table below

Confidence level z

     90%               1.64

     95%               1.96

     99%               2.58

Refer the z-table for the z-values.

Thus, a 99% confidence level is bigger than both the 95% and 90% confidence level.

Hence, a 95% confidence interval will be wider than the 90% confidence interval.