Question

1. a population has a mean of $mu = 50$ and a standard deviation of $sigma = 10$. a. if 3 points were added to every score in the population, what would be the new values for the mean and standard deviation? b. if every score in the population were multiplied by 2, then what would be the new values for the mean and standard deviation?

Explanation:

Step1: Recall mean and standard - deviation transformation rules

Let $\mu$ be the original mean and $\sigma$ be the original standard deviation. If we add a constant $c$ to each data - point $x_i$ in a population, the new mean $\mu_{new}$ and standard deviation $\sigma_{new}$ are given by: $\mu_{new}=\mu + c$ and $\sigma_{new}=\sigma$. If we multiply each data - point $x_i$ by a constant $k$, the new mean $\mu_{new}$ and standard deviation $\sigma_{new}$ are given by: $\mu_{new}=k\mu$ and $\sigma_{new}=k\sigma$.

Step2: Solve part (a)

Given $\mu = 50$, $\sigma = 10$, and $c = 3$.
For the mean: $\mu_{new}=\mu + c=50 + 3=53$.
For the standard deviation: $\sigma_{new}=\sigma = 10$.

Step3: Solve part (b)

Given $\mu = 50$, $\sigma = 10$, and $k = 2$.
For the mean: $\mu_{new}=k\mu=2\times50 = 100$.
For the standard deviation: $\sigma_{new}=k\sigma=2\times10 = 20$.

Answer:

a. Mean = 53, Standard deviation = 10
b. Mean = 100, Standard deviation = 20