Question

complete the following statements.
in general, % of the values in a data set lie at or below the median.
% of the values in a data set lie at or below the first quartile (q1).
if a sample consists of 1600 test scores, of them would be at or below the second quartile (q2).
if a sample consists of 1600 test scores, of them would be at or above the third quartile (q3).
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question 43
complete the following statements.
in general, % of the values in a data set lie at or below the 92th percentile.
% of the values in a data set lie at or above the 78th percentile..
if a sample consists of 700 test scores, of them would be at or below the 66th percentile.
if a sample consists of 700 test scores, of them would be at or above the 34th percentile.

Explanation:

Step1: Recall the definition of median

The median (Q2) divides the data set into two equal parts. So, 50% of the values in a data set lie at or below the median.

Step2: Recall the definition of first quartile

The first quartile (Q1) is the 25th percentile. So, 25% of the values in a data set lie at or below the first quartile.

Step3: Calculate the number of values at or below Q2

Since Q2 is the median (50th percentile), for a sample of \(n = 1600\) test scores, the number of values at or below Q2 is \(n\times0.5=1600\times0.5 = 800\).

Step4: Calculate the number of values at or above Q3

The third quartile (Q3) is the 75th percentile. So the proportion of values at or above Q3 is \(1 - 0.75=0.25\). For \(n = 1600\) test scores, the number of values at or above Q3 is \(n\times0.25=1600\times0.25 = 400\).

Step5: Recall the definition of percentile

By the definition of the \(k^{th}\) percentile, \(k\%\) of the values in a data set lie at or below the \(k^{th}\) percentile. So, 92% of the values in a data set lie at or below the 92nd percentile.

Step6: Calculate the proportion of values at or above a percentile

The proportion of values at or above the \(k^{th}\) percentile is \(100 - k\). For the 78th percentile, the proportion of values at or above is \(100 - 78=22\%\).

Step7: Calculate the number of values at or below a percentile

For the 66th percentile and \(n = 700\) test scores, the number of values at or below is \(n\times0.66=700\times0.66 = 462\).

Step8: Calculate the number of values at or above a percentile

The proportion of values at or above the 34th percentile is \(1 - 0.34 = 0.66\). For \(n=700\) test scores, the number of values at or above is \(n\times0.66=700\times0.66 = 462\).

Answer:

1. 50%
2. 25%
3. 800
4. 400
5. 92%
6. 22%
7. 462
8. 462