Question

question 46
for a sample with m = 10 and s = 2, a score of x = 13 would be considered an extreme score, far out in the tail of the distribution.
true
false
question 47
a population has μ = 30 and σ = 10. if 5 points are added to every score in the population, what are the new values for the mean and standard deviation?
μ = 40 and σ = 10
μ = 35 and σ = 15
μ = 35 and σ = 10
μ = 40 and σ = 15
question 48
calculate the variance and the standard deviation for the following sample data. scores: 1,0,3,1,2,4,0,5

Explanation:

Step1: Calculate z - score for Question 46

The formula for the z - score is $z=\frac{X - M}{s}$. Given $M = 10$, $s = 2$, and $X = 13$. Then $z=\frac{13 - 10}{2}=\frac{3}{2}=1.5$. In a normal distribution, extreme scores are typically considered to be $z> 2$ or $z < - 2$. Since $|z|=1.5<2$, the score is not extreme.

Step2: Analyze effect on mean and standard deviation for Question 47

If we add a constant $c$ to every score in a population, the new mean $\mu_{new}=\mu + c$ and the standard deviation $\sigma_{new}=\sigma$. Given $\mu = 30$, $\sigma = 10$, and $c = 5$. The new mean $\mu_{new}=30 + 5=35$ and the new standard - deviation $\sigma_{new}=10$.

Step3: Calculate mean for Question 48

The mean $\bar{X}$ of the data set $x_1,x_2,\cdots,x_n$ is $\bar{X}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 8$, and $\sum_{i=1}^{8}x_i=1 + 0+3 + 1+2 + 4+0 + 5=16$. So, $\bar{X}=\frac{16}{8}=2$.

Step4: Calculate variance for Question 48

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{X})^2}{n - 1}$.
$(1 - 2)^2=1$, $(0 - 2)^2 = 4$, $(3 - 2)^2=1$, $(1 - 2)^2=1$, $(2 - 2)^2=0$, $(4 - 2)^2=4$, $(0 - 2)^2=4$, $(5 - 2)^2=9$.
$\sum_{i = 1}^{8}(x_i - 2)^2=1+4 + 1+1+0+4+4+9 = 24$.
$s^{2}=\frac{24}{8 - 1}=\frac{24}{7}\approx3.43$.

Step5: Calculate standard deviation for Question 48

The standard deviation $s=\sqrt{s^{2}}=\sqrt{\frac{24}{7}}\approx1.85$.

Answer:

Question 46: B. False
Question 47: C. $\mu = 35$ and $\sigma = 10$
Question 48: Variance: $\frac{24}{7}\approx3.43$, Standard Deviation: $\sqrt{\frac{24}{7}}\approx1.85$