Question

given that ( z_{20} = -2 ) and ( z_{50} = -1 ), which of the following do you know?

• the variance is 10
• the standard deviation is 30
• the mean is 80
• the median is 40
• the data point ( x = 20 ) is 2 standard deviations from the mean
• the data point ( x = 50 ) is 1 standard deviation from the mean
• the data point ( x = 45 ) has a z - value of 1.5

done

Explanation:

Step1: Recall z-score formula

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

For $z_{20}=-2$, we have $-2=\frac{20 - \mu}{\sigma}$, which can be rewritten as $20-\mu=- 2\sigma$ (Equation 1).

For $z_{50}=-1$, we have $-1=\frac{50 - \mu}{\sigma}$, which can be rewritten as $50-\mu=-\sigma$ (Equation 2).

Step2: Solve for $\mu$ and $\sigma$

Subtract Equation 1 from Equation 2:

$$
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$$

Now substitute $\sigma = 30$ into Equation 2: $50-\mu=-30$, then $\mu=50 + 30=80$.

Step3: Analyze each option

• Variance is 10: Variance is $\sigma^{2}$. Since $\sigma = 30$, variance is $30^{2}=900

eq10$. So this is wrong.

• Standard deviation is 30: We found $\sigma = 30$, so this is correct.
• Mean is 80: We found $\mu = 80$, so this is correct.
• Median is 40: There is no information to determine the median from z - scores and two data points. So this is wrong.
• Data point $x = 20$ is 2 standard deviations from the mean: The z - score of $x = 20$ is $z_{20}=-2$, which means it is 2 standard deviations below the mean (since z - score is - 2). So this is correct.
• Data point $x = 50$ is 1 standard deviation from the mean: The z - score of $x = 50$ is $z_{50}=-1$, which means it is 1 standard deviation below the mean. So this is correct.
• Data point $x = 45$ has a z - value of 1.5: Using the z - score formula $z=\frac{45 - 80}{30}=\frac{-35}{30}\approx - 1.17

eq1.5$. So this is wrong.

Answer:

The standard deviation is 30, The mean is 80, The data point $x = 20$ is 2 standard deviations from the mean, The data point $x = 50$ is 1 standard deviation from the mean. (Options: "The standard deviation is 30", "The mean is 80", "The data point $x = 20$ is 2 standard deviations from the mean", "The data point $x = 50$ is 1 standard deviation from the mean")