Question

1. one population has a mean of μ = 50 and a standard deviation of σ = 15, and a different population has a mean of μ = 50 and a standard deviation of σ = 5. a. sketch both distributions, labelling μ and σ. b. would a score of x = 65 be considered an extreme value (out in the tail) in one of these distributions? explain your answer.

Explanation:

Step1: Sketch the normal - distribution for part a

For the first population with $\mu = 50$ and $\sigma=15$, the normal - distribution curve is centered at $x = 50$ with a relatively wide spread. For the second population with $\mu = 50$ and $\sigma = 5$, the normal - distribution curve is also centered at $x = 50$ but has a narrower spread. Mark the mean $\mu$ on the x - axis at the center of the curve and indicate the standard deviation $\sigma$ by marking points at $\mu\pm\sigma,\mu\pm2\sigma,\mu\pm3\sigma$ etc.

Step2: Calculate the z - score for part b

The z - score formula is $z=\frac{X-\mu}{\sigma}$. For the first population with $\mu = 50$, $\sigma = 15$ and $X = 65$, we have $z=\frac{65 - 50}{15}=\frac{15}{15}=1$. For the second population with $\mu = 50$, $\sigma = 5$ and $X = 65$, we have $z=\frac{65 - 50}{5}=\frac{15}{5}=3$.

Step3: Determine if the score is extreme

In a normal distribution, values with $|z|\geq3$ are often considered extreme (in the tails). For the first population with $z = 1$, the score of $X = 65$ is not extreme as it is within 1 standard - deviation of the mean. For the second population with $z = 3$, the score of $X = 65$ is extreme as it is 3 standard - deviations from the mean.

Answer:

a. Sketch two normal - distribution curves centered at $x = 50$, one with a wider spread ($\sigma = 15$) and one with a narrower spread ($\sigma = 5$), and label the mean $\mu = 50$ and standard deviation $\sigma$ on each.
b. For the population with $\mu = 50$ and $\sigma = 15$, a score of $X = 65$ is not extreme ($z = 1$). For the population with $\mu = 50$ and $\sigma = 5$, a score of $X = 65$ is extreme ($z = 3$).