given that $z_{20}=-2$ and $z_{50}=-1$, which of the following do you know?\nthe variance is 10.\nthe…

given that $z_{20}=-2$ and $z_{50}=-1$, which of the following do you know?\nthe variance is 10.\nthe standard deviation is 30.\nthe mean is 80.\nthe median is 40.\nthe data point $x = 20$ is 2 standard deviations from the mean.\nthe data point $x = 50$ is 1 standard deviation from the mean.\nthe data point $x = 45$ has a $z$-value of 1.5.

given that $z_{20}=-2$ and $z_{50}=-1$, which of the following do you know?\nthe variance is 10.\nthe standard deviation is 30.\nthe mean is 80.\nthe median is 40.\nthe data point $x = 20$ is 2 standard deviations from the mean.\nthe data point $x = 50$ is 1 standard deviation from the mean.\nthe data point $x = 45$ has a $z$-value of 1.5.

Answer

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. Given $z_{20}=-2$ and $z_{50}=-1$, we have $-2=\frac{20 - \mu}{\sigma}$ and $-1=\frac{50 - \mu}{\sigma}$.

Step2: Solve the system of equations

From $-2=\frac{20 - \mu}{\sigma}$, we get $-2\sigma=20 - \mu$, or $\mu=20 + 2\sigma$. From $-1=\frac{50 - \mu}{\sigma}$, we get $-\sigma=50 - \mu$, or $\mu=50+\sigma$. Then, set the two expressions for $\mu$ equal to each other: $20 + 2\sigma=50+\sigma$. Solving for $\sigma$ gives $\sigma = 30$. Substitute $\sigma = 30$ into $\mu=50+\sigma$ to get $\mu=80$.

Step3: Analyze each option

  • Variance: Since $\sigma = 30$, variance $\sigma^{2}=900\neq10$.
  • Standard deviation: $\sigma = 30$.
  • Mean: $\mu = 80$.
  • Median: We have no information about the shape of the distribution, so we cannot determine the median.
  • For $x = 20$, $z=\frac{20 - 80}{30}=\frac{-60}{30}=-2$, so $x = 20$ is 2 standard deviations from the mean.
  • For $x = 50$, $z=\frac{50 - 80}{30}=\frac{-30}{30}=-1$, so $x = 50$ is 1 standard deviation from the mean.
  • For $x = 45$, $z=\frac{45 - 80}{30}=\frac{-35}{30}\neq1.5$.

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.