here are the summary statistics for the all - time top 50 womens long jump distances (in meters).
|n|mean|sd|min|q1|med|q3|max|
|50|7.2|0.12|7.06|7.11|7.17|7.27|7.52|
1. calculate the z - score for american jackie joyner - kersee, who jumped 7.49 meters in 1994.
2. how far did jackie joyner - kersee jump in feet (1 meter = 3.28 feet)?
3. if all 50 jumps were converted to feet, what would be the new mean and new standard deviation of the distances?
4. calculate the z - score for jackie joyner - kersees jump when it is measured in feet. what do you notice?
a local farmer sells chicken eggs at a farmers market every weekend. she keeps track of the number of dozens of eggs sold each weekend for an entire year, and finds the mean is 14.7 dozen and the standard deviation is 3.0 dozen.
1. what is the variance for the distribution of dozens of eggs sold?
2. suppose the farmer converts all the values from dozens of eggs to number of eggs. what is the new mean and new standard deviation?
3. what is the variance for the distribution of number of eggs sold?
4. how does this new variance compare to the original variance?

Question

Accepted Answer

### 1. Calculate the z - score for American Jackie Joyner - Kersee who jumped 7.49 meters in 1994
# Explanation:
## Step1: Recall z - score formula
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Given $\mu = 7.2$, $\sigma=0.12$, and $x = 7.49$.
## Step2: Substitute values into formula
$z=\frac{7.49 - 7.2}{0.12}=\frac{0.29}{0.12}\approx2.42$
# Answer:
$z\approx2.42$

### 2. How far did Jackie Joyner - Kersee jump in feet (1 meter = 3.28 feet)?
# Explanation:
## Step1: Use conversion factor
Multiply the distance in meters by the conversion factor.
$d = 7.49	imes3.28$
## Step2: Calculate the result
$d=7.49	imes3.28 = 24.5672$ feet
# Answer:
$24.5672$ feet

### 3. If all 50 jumps were converted to feet, what would be the new mean and new standard deviation of the distances?
# Explanation:
## Step1: Convert the mean
Since we multiply each data - point by 3.28 to convert from meters to feet, the new mean $\mu_{new}=7.2	imes3.28 = 23.616$ feet.
## Step2: Convert the standard deviation
The standard deviation also gets multiplied by the conversion factor. So $\sigma_{new}=0.12	imes3.28 = 0.3936$ feet.
# Answer:
New mean: $23.616$ feet, New standard deviation: $0.3936$ feet

### 4. Calculate the z - score for Jackie Joyner - Kersee's jump when it is measured in feet. What do you notice?
# Explanation:
## Step1: Recall the new mean and standard deviation in feet
We found $\mu_{new}=23.616$ feet and $\sigma_{new}=0.3936$ feet, and $x_{new}=24.5672$ feet.
## Step2: Calculate the z - score
$z=\frac{24.5672 - 23.616}{0.3936}=\frac{0.9512}{0.3936}\approx2.42$
We notice that the z - score is the same as when the measurement was in meters. This is because the z - score is a unit - less measure that standardizes the data point relative to the mean and standard deviation of the distribution, and multiplying all data points by a constant does not change the relative position of a particular data point within the distribution.
# Answer:
$z\approx2.42$, and the z - score is the same as when measured in meters.

### 5. What is the variance for the distribution of dozens of eggs sold?
# Explanation:
## Step1: Recall the relationship between variance and standard deviation
The variance $	ext{Var}(X)=\sigma^{2}$, where $\sigma$ is the standard deviation.
Given $\sigma = 3.0$ dozen.
## Step2: Calculate the variance
$	ext{Var}(X)=3.0^{2}=9.0$ (dozen$^{2}$)
# Answer:
$9.0$ (dozen$^{2}$)

### 6. Suppose the farmer converts all the values from dozens of eggs to number of eggs. What is the new mean and new standard deviation?
# Explanation:
## Step1: Recall the conversion factor
Since 1 dozen = 12 eggs, to convert from dozens to number of eggs, we multiply each data - point by 12.
The original mean is $\mu = 14.7$ dozen. The new mean $\mu_{new}=14.7	imes12=176.4$ eggs.
The original standard deviation is $\sigma = 3.0$ dozen. The new standard deviation $\sigma_{new}=3.0	imes12 = 36$ eggs.
# Answer:
New mean: $176.4$ eggs, New standard deviation: $36$ eggs

### 7. What is the variance for the distribution of number of eggs sold?
# Explanation:
## Step1: Recall the new standard deviation in eggs
The new standard deviation $\sigma_{new}=36$ eggs.
## Step2: Calculate the variance
Since $	ext{Var}(X)=\sigma^{2}$, the new variance $	ext{Var}_{new}=36^{2}=1296$ (eggs$^{2}$)
# Answer:
$1296$ (eggs$^{2}$)

### 8. How does this new variance compare to the original variance?
# Explanation:
## Step1: Recall the original and new variances
The original variance was 9.0 (dozen$^{2}$). The new variance is 1296 (eggs$^{2}$).
Since we multiplied each data - point by 12 to convert from dozens to eggs, and variance is the square of the standard deviation, the new variance is $12^{2}$ times the original variance. $9	imes12^{2}=9	imes144 = 1296$.
# Answer:
The new variance is $144$ times the original variance.

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QUESTION IMAGE

Question

Explanation:

1. Calculate the z - score for American Jackie Joyner - Kersee who jumped 7.49 meters in 1994

Step1: Recall z - score formula

Step2: Substitute values into formula

Step1: Use conversion factor

Step2: Calculate the result

Step1: Convert the mean

Step2: Convert the standard deviation

Answer:

2. How far did Jackie Joyner - Kersee jump in feet (1 meter = 3.28 feet)?

n	mean	sd	min	q1	med	q3	max
50	7.2	0.12	7.06	7.11	7.17	7.27	7.52