Suppose that, based on a simple random sample of n = 100 American adult men, we wish to calculate a 98% confidence interval for the population mean height µ. For this confidence interval, the critical values ± z* could be calculated from:
the 1st percentile of a N(0, 1) distribution.
the 2nd percentile of a N(0, 1) distribution.
the 2.5th percentile of a N(0, 1) distribution.
the 98th percentile of a N(0, 1) distribution.
The correct answer is: the 2.5th percentile of a N(0, 1) distribution.
Why would the 1st percentile not be used?
The 1st percentile corresponds to a very low value in the distribution. In the context of constructing a confidence interval, it would represent an extremely small or negative value. Since height cannot have negative values and we are looking to estimate the population mean height, using the 1st percentile would not be appropriate for calculating the critical values.
why would it be 2.5 percentile and not 99 percentile
Apologies for the confusion in my previous response. The correct answer is the 99th percentile of a N(0, 1) distribution, not the 2.5th percentile.
To calculate a confidence interval, we need to determine the critical values that define the range of values within which the population parameter (in this case, the population mean height) is likely to fall.
In a normal distribution, which is assumed in this case, the middle 98% of the data lies within two standard deviations of the mean. Therefore, to capture 98% of the distribution, we need to find the value that is 2 standard deviations away from the mean on either side.
Since the standard normal distribution (N(0, 1)) has a mean of 0 and a standard deviation of 1, the critical values for a 98% confidence interval would be located at the 99th percentile. This means that 99% of the distribution is below that value, leaving the remaining 1% in the tails (0.5% in each tail), which represents the level of significance for the confidence interval (α = 0.02 or 2%). Therefore, the critical values ± z* can be calculated from the 99th percentile of a N(0, 1) distribution.
To calculate a confidence interval, we need critical values, which represent the number of standard deviations from the mean. In this case, we need a 98% confidence interval for the population mean height.
First, we need to find the area in the tails of the standard normal distribution. Since it's a 98% confidence interval, we divide this area into two equal parts: 1% on each tail.
To find the critical values for each tail, we need to find the z-score that corresponds to the area in each tail. The standard normal distribution has a mean of 0 and a standard deviation of 1, denoted as N(0, 1).
Now let's determine which option is correct:
1. The 1st percentile of a N(0, 1) distribution.
This would correspond to finding the z-score that leaves 1% in one tail. However, we need 2% in the tails for a 98% confidence interval, so this option is incorrect.
2. The 2nd percentile of a N(0, 1) distribution.
Similar to option 1, this would correspond to finding the z-score that leaves 2% in one tail. However, we need 2% in each tail for a 98% confidence interval, so this option is also incorrect.
3. The 2.5th percentile of a N(0, 1) distribution.
This corresponds to finding the z-score that leaves 2.5% in one tail. Since we need 2% in each tail for a 98% confidence interval, this option is correct. The critical value for this option would be denoted as z*.
4. The 98th percentile of a N(0, 1) distribution.
This would correspond to finding the z-score that leaves 98% in one tail. However, we need 2% in each tail for a 98% confidence interval, so this option is incorrect.
Therefore, the correct answer is "the 2.5th percentile of a N(0, 1) distribution."