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

When constructing a 95% confidence interval (compared to a 90% interval) with the same data, the interval becomes wider and the probability of making an error by not capturing the true population mean decreases. Higher confidence levels result in wider intervals as they aim to include more of the distribution to encapsulate the true mean with greater certainty.

Step-by-step explanation:

If a 90% confidence interval is constructed for the population mean, and then a 95% confidence interval is constructed instead with everything else remaining the same, the width of the interval would be greater, and the probability of making an error would be decreased. The reason for this is because higher confidence levels require wider intervals to ensure that the true population mean lies within the interval with greater certainty. For example, if the 90% confidence interval is (67.18, 68.82), then a 95% confidence interval might be (67.02, 68.98), which is indeed wider.

As the confidence level increases, the error bound for the confidence interval increases because you need more area under the normal distribution curve to capture the true population mean. This means that the 95% confidence interval will be wider than the 90% confidence interval because it tries to include more of the distribution to achieve the higher confidence. Conversely, if the confidence level were to decrease to say 90% for the same data set, the error bound would decrease along with it.

Summing up, when we say we are 90% confident in constructing a confidence interval for a mean, we imply that if we took repeated samples and constructed confidence intervals from those samples, approximately 90% of the intervals would contain the true value of the population mean.

Answer 2

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* CV* (\sigma)/(√(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.