Question

Recall from Chapter 7 that the interquartile range (IQR) covers the middle $50 \%$ of the data. a. What range of standardized scores is covered by the interquartile range for normally distributed populations? In other words, what are the standardized scores for the lower quartile (the $25^{\operatorname{th}}$ percentile) and the upper quartile (the $75^{\text {th }}$ percentile)? (Hint: Draw a standard normal curve and locate the 25 th and 75 th percentiles using Table $8.1 .$ ) b. How many standard deviations are covered by the interquartile range for a normally distributed population? (Hint: Use the standardized scores found in part (a) for the lower and upper quartiles.) c. Data values are outliers for normally distributed data if they are more than two IQRs away from the mean. (See Exercise 14.) At what percentiles (at the upper and lower ends) are data values considered outliers for normally distributed data? Use your answers to parts (a) and (b) to help you determine the answer to this question. (Software required.)

          Recall from Chapter 7 that the interquartile range (IQR) covers the middle $50 \%$ of the data.
a. What range of standardized scores is covered by the interquartile range for normally distributed populations? In other words, what are the standardized scores for the lower quartile (the $25^{\operatorname{th}}$ percentile) and the upper quartile (the $75^{\text {th }}$ percentile)? (Hint: Draw a standard normal curve and locate the 25 th and 75 th percentiles using Table $8.1 .$ )
b. How many standard deviations are covered by the interquartile range for a normally distributed population? (Hint: Use the standardized scores found in part (a) for the lower and upper quartiles.)
c. Data values are outliers for normally distributed data if they are more than two IQRs away
from the mean. (See Exercise 14.) At what percentiles (at the upper and lower ends) are data values considered outliers for normally distributed data? Use your answers to parts (a) and
(b) to help you determine the answer to this question. (Software required.)

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Allan G. Bluman 9th Edition

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Solved by Expert Avi Zellman

03/04/2022

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The standardized scores for the lower quartile (the 25th percentile) and the upper quartile (the 75th percentile) in a standard normal distribution are approximately -0.67 and 0.67, respectively. This is because the 25th percentile corresponds to a z-score of Show more…

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Recall from Chapter 7 that the interquartile range (IQR) covers the middle $50 \%$ of the data. a. What range of standardized scores is covered by the interquartile range for normally distributed populations? In other words, what are the standardized scores for the lower quartile (the $25^{\operatorname{th}}$ percentile) and the upper quartile (the $75^{\text {th }}$ percentile)? (Hint: Draw a standard normal curve and locate the 25 th and 75 th percentiles using Table $8.1 .$ ) b. How many standard deviations are covered by the interquartile range for a normally distributed population? (Hint: Use the standardized scores found in part (a) for the lower and upper quartiles.) c. Data values are outliers for normally distributed data if they are more than two IQRs away from the mean. (See Exercise 14.) At what percentiles (at the upper and lower ends) are data values considered outliers for normally distributed data? Use your answers to parts (a) and (b) to help you determine the answer to this question. (Software required.)

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Transcript

00:01 We're looking at a question focused on the inner quartile range, i qr.

00:06 And the inner quartile range, remember, is like that middle 50 % of data.

00:13 And there's a couple pieces here that i don't have access to, like different tables from the question of things like that.

00:20 But it's helpful just to look at the inner quartile range as a normalized curve.

00:27 So it'll look something like this.

00:34 And again, this inner quartile range, you know, covers the middle 50%.

00:38 So if you picture this as being the median, the inner quartile range is everywhere in between 25 percentile and the 75th percentile.

00:51 So this whole section here would be 50 percent of the overall information.

00:58 So our first question is, what are the standardized scores for the lower quartile, 25%, and the upper quartile, 75%.

01:17 And we want to think about this in terms of like their relationship to standard deviations.

01:27 So our standard deviation covers a total of, 68 % of that of like the similar space.

01:38 So it kind of goes out further.

01:40 It goes out to about here.

01:42 And that whole distance is 68 .27 % of the total data, which would be one standard deviation to the left of the median and one standard deviation to the right of the medium.

02:00 And so what we find is that the percentage, like the amount of that, the inner, the lower quartile and the upper quartile each are away from the median is equal to 0 .6745 of the standard deviation.

02:25 And so on the left, it would be negative 0 .6745 of the median.

02:36 So that's our distance away, or what we could say is our standardized scores of the upper and lower quartile.

02:44 So when we think about the standard deviations covered by the inner quartile range, what we really want to do is just add them, the absolute values at least.

02:52 So we have 0 .6745.

02:55 Four or five standard deviations to the left and 0 .6745 standard deviations to the right.

03:07 And so together it's 1 .349 standard deviations.

03:16 And that would be the total amount that the inner quartile range would cover.

03:24 So sort of like that entire area under the curve...

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