Question

1. the first test in introductory algebra had a mean of 82 with a standard deviation of 15. if a student made a 75 on the test, what is the corresponding z - score?

a. - 0.58 b. - 0.47 c. 0.47 d. 0.82

1. what is the interquartile range for the following data? 4, 5, 7, 8, 9, 10

a. 5 b. 9 c. 4 d. 6

1. using the data set of: 10, 12, 13, 15, 18, 20, 25, 27, 32, 36, 39, and 40, what is the interquartile range?

a. 20 b. 19 c. 21 d. 18

1. what is the standard deviation of the following probability distribution?

x 0 2 4 6 8
p(x) 0.25 0.1 0.3 0.25 0.1
a. 4.4 b. 2.6 c. 3.7 d. 6.9

1. the number of cartoons watched by first - graders on saturday mornings is shown below:

x 0 1 2 3 4 5
p(x) 0.2 0.2 0.3 0.15 0.1 0.05
what is the mean of the distribution?
a. 1.9 b. 2.6 c. 2.4 d. 3.1

1. a probability distribution is constructed for the number of boys a family with two children has. let “x” be the number of boys. what is the probability for x = 2?

a. 1/4 b. 1/2 c. 1/3 d. 3/4

1. what value would be needed to complete the following distribution?

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Identify values

Given $\mu = 82$, $\sigma=15$, and $x = 75$.

Step3: Calculate z - score

$z=\frac{75 - 82}{15}=\frac{-7}{15}\approx - 0.47$

Step1: Find median for lower and upper half

For the data set $4,5,7,8,9,10$, first find the median of the whole set. Since $n = 6$ (even), the median is the average of the 3rd and 4th ordered values, $M=\frac{7 + 8}{2}=7.5$.
The lower half is $4,5,7$ and its median $Q_1 = 5$. The upper half is $8,9,10$ and its median $Q_3=9$.

Step2: Calculate inter - quartile range

The inter - quartile range $IQR=Q_3 - Q_1=9 - 5 = 4$

Step1: Find median for lower and upper half

For the data set $10,12,13,15,18,20,25,27,32,36,39,40$ with $n = 12$ (even). The median of the whole set is the average of the 6th and 7th ordered values, $M=\frac{20 + 25}{2}=22.5$.
The lower half is $10,12,13,15,18,20$ and its median $Q_1=\frac{13+15}{2}=14$. The upper half is $25,27,32,36,39,40$ and its median $Q_3=\frac{32 + 36}{2}=34$.

Step2: Calculate inter - quartile range

$IQR=Q_3 - Q_1=34 - 14=20$

Step1: Calculate the mean $\mu$

$\mu=\sum_{i}x_iP(x_i)=0\times0.25 + 2\times0.1+4\times0.3+6\times0.25+8\times0.1=0 + 0.2+1.2 + 1.5+0.8=3.7$

Step2: Calculate the variance $\sigma^{2}$

$\sigma^{2}=\sum_{i}(x_i-\mu)^2P(x_i)=(0 - 3.7)^2\times0.25+(2 - 3.7)^2\times0.1+(4 - 3.7)^2\times0.3+(6 - 3.7)^2\times0.25+(8 - 3.7)^2\times0.1$
$=( - 3.7)^2\times0.25+( - 1.7)^2\times0.1+(0.3)^2\times0.3+(2.3)^2\times0.25+(4.3)^2\times0.1$
$=13.69\times0.25 + 2.89\times0.1+0.09\times0.3+5.29\times0.25+18.49\times0.1$
$=3.4225+0.289+0.027+1.3225+1.849 = 6.91$

Step3: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{\sigma^{2}}=\sqrt{6.91}\approx2.6$

Answer:

b. - 0.47