7. A researcher selects a random sample. A 90% confidence interval for a population mean


A) is an interval with margin of error ± 90%. B) has the property that if we repeatedly selected our random sample in exactly the same
way, each time constructing a different 90% confidence interval for , then in the
long run 90% of those intervals would contain . C) (a) and (b) are both true. D) is an interval that has width .90.

The correct answer is B) The 90% confidence interval has the property that if we repeatedly selected our random sample in exactly the same way, each time constructing a different 90% confidence interval for the population mean μ, then in the long run 90% of those intervals would contain μ.

To understand why option B) is correct, let's elaborate on the concept of a confidence interval and how it is calculated:

A confidence interval is a range of values within which a population parameter, such as the population mean, is estimated to lie with a certain level of confidence. In this case, we want to estimate the population mean with a 90% confidence level.

When constructing a confidence interval, we use sample data to estimate the population parameter. The margin of error, which is the maximum difference between the sample mean and the population mean, is determined based on the variability of the data and the desired level of confidence.

The formula for calculating the confidence interval is:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Error)

The critical value is obtained from the standardized normal distribution (Z-distribution) or the t-distribution, depending on the sample size and whether the population standard deviation is known. The standard error is a measure of the variability of the sample mean.

Now, let's consider option B):

If we repeatedly select random samples from the same population, construct 90% confidence intervals for the population mean using different samples, and calculate the long-run proportion of intervals that contain the true population mean, we expect this proportion to be around 90% (assuming our sampling methods are unbiased and our assumptions are met). This is what is meant by the statement in option B) - "in the long run, 90% of those intervals would contain μ."

Option A) is incorrect because a 90% confidence interval does not have a fixed margin of error ±90%. The margin of error varies depending on the sample data and the desired level of confidence.

Option C) is incorrect because only statement B) is true. Option D) is incorrect because the width of a confidence interval does not directly correspond to the confidence level.

The correct answer is B) has the property that if we repeatedly selected our random sample in exactly the same way, each time constructing a different 90% confidence interval for μ, then in the long run 90% of those intervals would contain μ.

A 90% confidence interval is not an interval with a margin of error of ±90% (option A). The margin of error corresponds to the precision or range of values around the sample estimate within which the population parameter is expected to lie.

Option D is incorrect because a 90% confidence interval does not have a width of 0.90. The width of a confidence interval depends on the sample data and the variability of the population.

Option C is incorrect because only statement B is true.

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