Answer :
To solve the problem, we need to determine the correct information about the given data based on the z-scores provided.
Given:
- [tex]\( z_{20} = -2 \)[/tex]
- [tex]\( z_{50} = -1 \)[/tex]
### Step 1: Set up the equations for z-scores
The z-score formula is given by:
[tex]\[ z = \frac{(x - \mu)}{\sigma} \][/tex]
where [tex]\( \mu \)[/tex] is the mean and [tex]\( \sigma \)[/tex] is the standard deviation.
Using the given z-scores:
[tex]\[ z_{20} = -2 \rightarrow -2 = \frac{(20 - \mu)}{\sigma} \][/tex]
[tex]\[ z_{50} = -1 \rightarrow -1 = \frac{(50 - \mu)}{\sigma} \][/tex]
### Step 2: Solve for the mean [tex]\(\mu\)[/tex]
Rewrite the equations in terms of [tex]\(\mu\)[/tex] and [tex]\(\sigma\)[/tex]:
[tex]\[ -2 = \frac{(20 - \mu)}{\sigma} \rightarrow 20 - \mu = -2\sigma \rightarrow \mu = 20 + 2\sigma \][/tex]
[tex]\[ -1 = \frac{(50 - \mu)}{\sigma} \rightarrow 50 - \mu = -\sigma \rightarrow \mu = 50 + \sigma \][/tex]
Equating the two expressions for [tex]\(\mu\)[/tex]:
[tex]\[ 20 + 2\sigma = 50 + \sigma \][/tex]
Solving for [tex]\(\sigma\)[/tex]:
[tex]\[ 20 + 2\sigma = 50 + \sigma \][/tex]
[tex]\[ 2\sigma - \sigma = 50 - 20 \][/tex]
[tex]\[ \sigma = 30 \][/tex]
Substitute [tex]\(\sigma\)[/tex] back into one of the expressions for [tex]\(\mu\)[/tex]:
[tex]\[ \mu = 20 + 2\sigma = 20 + 2(30) = 20 + 60 = 80 \][/tex]
### Step 3: Verify the variance
The variance ([tex]\(\sigma^2\)[/tex]) is given by:
[tex]\[ \sigma^2 = 30^2 = 900 \][/tex]
### Step 4: Since the median is already given
The median is given as [tex]\(40\)[/tex].
### Step 5: Analyze the options
- "The variance is 10": This is incorrect as the variance is [tex]\(900\)[/tex].
- "The standard deviation is 30": This is correct as we calculated [tex]\(\sigma = 30\)[/tex].
- "The mean is 80": This is correct as we calculated [tex]\(\mu = 80\)[/tex].
- "The median is 40": This is correct as it is given.
- "The data point [tex]\(x = 20\)[/tex] is 2 standard deviations from the mean": This is incorrect because the deviation is correct but the calculation does not match [tex]\(z_{20}\)[/tex].
- "The data point [tex]\(x = 50\)[/tex] is 1 standard deviation from the mean": This is correct as calculated.
- "The data point [tex]\(x = 45\)[/tex] has a [tex]\(z\)[/tex]-value of 1.5": This is incorrect based on our earlier calculations.
### Conclusion
The correct answers are:
- The standard deviation is 30.
- The mean is 80.
- The median is 40.
- The data point [tex]\( x = 50 \)[/tex] is 1 standard deviation from the mean.
Thus, the valid responses are:
"The standard deviation is 30."
"The mean is 80."
"The median is 40."
"The data point [tex]\( x = 50 \)[/tex] is 1 standard deviation from the mean."
Given:
- [tex]\( z_{20} = -2 \)[/tex]
- [tex]\( z_{50} = -1 \)[/tex]
### Step 1: Set up the equations for z-scores
The z-score formula is given by:
[tex]\[ z = \frac{(x - \mu)}{\sigma} \][/tex]
where [tex]\( \mu \)[/tex] is the mean and [tex]\( \sigma \)[/tex] is the standard deviation.
Using the given z-scores:
[tex]\[ z_{20} = -2 \rightarrow -2 = \frac{(20 - \mu)}{\sigma} \][/tex]
[tex]\[ z_{50} = -1 \rightarrow -1 = \frac{(50 - \mu)}{\sigma} \][/tex]
### Step 2: Solve for the mean [tex]\(\mu\)[/tex]
Rewrite the equations in terms of [tex]\(\mu\)[/tex] and [tex]\(\sigma\)[/tex]:
[tex]\[ -2 = \frac{(20 - \mu)}{\sigma} \rightarrow 20 - \mu = -2\sigma \rightarrow \mu = 20 + 2\sigma \][/tex]
[tex]\[ -1 = \frac{(50 - \mu)}{\sigma} \rightarrow 50 - \mu = -\sigma \rightarrow \mu = 50 + \sigma \][/tex]
Equating the two expressions for [tex]\(\mu\)[/tex]:
[tex]\[ 20 + 2\sigma = 50 + \sigma \][/tex]
Solving for [tex]\(\sigma\)[/tex]:
[tex]\[ 20 + 2\sigma = 50 + \sigma \][/tex]
[tex]\[ 2\sigma - \sigma = 50 - 20 \][/tex]
[tex]\[ \sigma = 30 \][/tex]
Substitute [tex]\(\sigma\)[/tex] back into one of the expressions for [tex]\(\mu\)[/tex]:
[tex]\[ \mu = 20 + 2\sigma = 20 + 2(30) = 20 + 60 = 80 \][/tex]
### Step 3: Verify the variance
The variance ([tex]\(\sigma^2\)[/tex]) is given by:
[tex]\[ \sigma^2 = 30^2 = 900 \][/tex]
### Step 4: Since the median is already given
The median is given as [tex]\(40\)[/tex].
### Step 5: Analyze the options
- "The variance is 10": This is incorrect as the variance is [tex]\(900\)[/tex].
- "The standard deviation is 30": This is correct as we calculated [tex]\(\sigma = 30\)[/tex].
- "The mean is 80": This is correct as we calculated [tex]\(\mu = 80\)[/tex].
- "The median is 40": This is correct as it is given.
- "The data point [tex]\(x = 20\)[/tex] is 2 standard deviations from the mean": This is incorrect because the deviation is correct but the calculation does not match [tex]\(z_{20}\)[/tex].
- "The data point [tex]\(x = 50\)[/tex] is 1 standard deviation from the mean": This is correct as calculated.
- "The data point [tex]\(x = 45\)[/tex] has a [tex]\(z\)[/tex]-value of 1.5": This is incorrect based on our earlier calculations.
### Conclusion
The correct answers are:
- The standard deviation is 30.
- The mean is 80.
- The median is 40.
- The data point [tex]\( x = 50 \)[/tex] is 1 standard deviation from the mean.
Thus, the valid responses are:
"The standard deviation is 30."
"The mean is 80."
"The median is 40."
"The data point [tex]\( x = 50 \)[/tex] is 1 standard deviation from the mean."
