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Recall that the study reported that with 95% confidence, the underlying mean number of drinks consumed by Australian university students on a typical drinking day is somewhere between 4.8 and 6. Around 100 responses from Australian university students were used to construct this confidence interval.If the study had instead involved around 400 responses, given that the sample standard deviation did not change, the 95% confidence interval for the underlying mean number of drinks consumed by Australian university students on a typical drinking day would have been:about 1/2 as wideabout 1/4 as widepretty much unchangedabout 4 times wider

Question

Recall that the study reported that with 95% confidence, the underlying mean number of drinks consumed by Australian university students on a typical drinking day is somewhere between 4.8 and 6. Around 100 responses from Australian university students were used to construct this confidence interval.If the study had instead involved around 400 responses, given that the sample standard deviation did not change, the 95% confidence interval for the underlying mean number of drinks consumed by Australian university students on a typical drinking day would have been:about 1/2 as wideabout 1/4 as widepretty much unchangedabout 4 times wider

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Solution

The 95% confidence interval for the underlying mean number of drinks consumed by Australian university students on a typical drinking day would have been about 1/2 as wide.

Here's why:

The width of a confidence interval is determined by the standard error of the mean, which is the standard deviation divided by the square root of the sample size. If the sample size increases by a factor of 4 (from 100 to 400), the standard error of the mean decreases by a factor of 2 (since the square root of 4 is 2).

Therefore, if the standard deviation remains the same, and the sample size quadruples, the standard error of the mean (and thus the width of the confidence interval) would be halved.

This problem has been solved

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