Question

questions 4 and 5 refer to the following setting. forty students took a statistics exam having a maximum of 50 points. the distribution of score is given in the stem - and - leaf plot.

1. which of the following are the correct median and interquartile range of this distribution?

a. median = 31; iqr = 21
c. median = 32; iqr = 23 to 44
b. median = 32; iqr = 21
d. median = 33; iqr = 21

1. the standard deviation of this distribution of exam scores is about 13. which of the following is a correct interpretation of this value?

a. every students exam score is within 13 points of the mean.
b. the gap between the highest and lowest score on the exam is 13 points.
c. the exam scores typically vary from each other by 13 points.
d. the exam scores typically vary from the mean by 13 points.
questions 6 and 7 refer to the following setting. in a statistics class with 136 students, the professor records how much money (in dollars) each student has in his or her possession during the first class of the semester. the histogram shows the data that were collected.

1. the percentage of students with less than $10 in their possession is closest to

a. 30%
c. 45%
b. 35%
d. 60%

1. which of the following statements about this distribution can be concluded from the graph?

a. the distribution is left - skewed.
b. the median is between $10 and $20.
c. the standard deviation of the distribution is more than $60.
d. the mean is likely less than the median.

Explanation:

Question 4

Step1: Find the median

There are \(n = 40\) data points. The median is the average of the \(20^{th}\) and \(21^{st}\) data points.
Counting the leaves:

• For stem \(0\): \(2\) leaves
• For stem \(1\): \(4\) leaves
• For stem \(2\): \(10\) leaves
• For stem \(3\): \(8\) leaves

The \(20^{th}\) and \(21^{st}\) data points are in stem \(3\). The \(20^{th}\) data point is \(31\) and the \(21^{st}\) data point is \(33\). The median \(M=\frac{31 + 33}{2}=32\)

Step2: Find \(Q_1\) and \(Q_3\)

The lower half of the data (\(n_1=20\) data points) has the median \(Q_1\). The \(10^{th}\) and \(11^{th}\) data points of the lower - half.
Counting the leaves of the lower - half (stems \(0\), \(1\), \(2\)):

• For stem \(0\): \(2\) leaves
• For stem \(1\): \(4\) leaves
• For stem \(2\): \(10\) leaves. The \(10^{th}\) and \(11^{th}\) data points of the lower - half are \(23\) and \(23\), so \(Q_1 = 23\)

The upper half of the data (\(n_2 = 20\) data points) has the median \(Q_3\). The \(10^{th}\) and \(11^{th}\) data points of the upper - half (stems \(3\), \(4\), \(5\)):

• For stem \(3\): \(8\) leaves
• For stem \(4\): \(10\) leaves. The \(10^{th}\) and \(11^{th}\) data points of the upper - half are \(44\) and \(44\), so \(Q_3=44\)

The inter - quartile range \(IQR=Q_3 - Q_1=44 - 23 = 21\)

The standard deviation is a measure of the spread of a data set. It represents the typical distance of the data points from the mean.

• Option a: Not every data point is within \(1\) standard deviation of the mean. In a normal distribution, about \(68\%\) of data is within \(1\) standard deviation of the mean, but this is not a strict rule for all distributions.
• Option b: The range (highest - lowest) is a different measure of spread. The standard deviation is not related to the range in this way.
• Option c: The standard deviation is not about the spread between data points themselves but about their spread from the mean.
• Option d: The standard deviation of \(13\) means that exam scores typically vary from the mean by \(13\) points.

Step1: Estimate the number of students with less than \(\$10\)

The total number of students \(n = 136\). The frequency of the first bar (less than \(\$10\)) is approximately \(60\)

Step2: Calculate the percentage

The percentage \(P=\frac{60}{136}\times100\approx44.12\%\)

Answer:

b. Median \( = 32\); \(IQR = 21\)

Question 5