9. Properties of a confidence interval AaAa? Suppose the mean of a population is

Question

9. Properties of a confidence interval AaAa? Suppose the mean of a population is 22. A researcher (who does not know that ?-22) selects a random sample of size n from this population. Then he constructs a 95% confidence interval of the population mean. The true population mean and the researchers 95% confidence interval of the population mean are shown in the following graph. Use the graph to answer the questions that follow. Sample Mean 95% confidence interva of the Population Mean True Population Mean 21 22 23 24 25 26 27 28 29 30 31 Use the grey star to mark the mean of the sample. (Be sure to place the star on the horizontal blue line segment that represents the confidence interval.) To construct the confidence interval, the quantity tsn is subtracted from and added to the sample mean. In this case, ts Suppose that the researcher uses the same sample but decides to construct a 85% confidence interval of the mean of the population. Compared with the center of the 95% confidence interval, the center of the 85% confidence interval . Compared to the 95% confidence interval, the 85% confidence interval At the same confidence level, confidence interval as changing the confidence level from 9S96 to 85%. in the sample size would have a similar effect on the width of the

Explanation / Answer

Sol:

a)

The grey star (sample mean) must be midway between the confidence interval, that is between 23 and 30.

Hence, the grey star has to be placed at = 26.5 [ANSWER]

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b)

Here,

t*sM = upper bound - sample mean = 30 - 26.5 = 3.5 [ANSWER]

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c)

Compared with the center of the 95% confidence interval, the center of the 85% confidence interval [IS THE SAME.]

Compared to the 95% confidence interval, the 85% confidence interval is [NARROWER.]

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d)

At the same confidence level, [INCREASING] in the sample size would have a similar effect on the width of the confidence interval as changing the confidence level from 95% to 85%.

This is so because increasing confidence levels makes the interval wider, and so is decreasing sample size.

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