(a) Compute the z-score corresponding to the individual who obtained 35.2 miles per gallon. Interpret this result.
The z-score corresponding to the individual is --- and indicates that the data value is ---standard deviation(s)
--- the mean.
data set:
32.3
35.9
37.8
38.6
39.9
42.5
34.6
36.3
37.9
38.7
40.6
42.8
34.8
37.3
38.2
39.0
41.4
43.3
35.2
37.5
38.4
39.7
41.8
49.0
To compute the z-score, we need to find the mean and standard deviation of the given data set.
Mean: To calculate the mean, we sum up all the values and divide by the number of data points.
Mean = (32.3 + 35.9 + 37.8 + 38.6 + 39.9 + 42.5 + 34.6 + 36.3 + 37.9 + 38.7 + 40.6 + 42.8 + 34.8 + 37.3 + 38.2 + 39.0 + 41.4 + 43.3 + 35.2 + 37.5 + 38.4 + 39.7 + 41.8 + 49.0) / 24 = 38.879
Standard Deviation: To calculate the standard deviation, we need to find the variance first. Variance is calculated by finding the difference between each data point and the mean, squaring it, summing up these squared differences, and then dividing it by the number of data points.
Variance = [(32.3 - 38.879)^2 + (35.9 - 38.879)^2 + (37.8 - 38.879)^2 + (38.6 - 38.879)^2 + (39.9 - 38.879)^2 + (42.5 - 38.879)^2 + (34.6 - 38.879)^2 + (36.3 - 38.879)^2 + (37.9 - 38.879)^2 + (38.7 - 38.879)^2 + (40.6 - 38.879)^2 + (42.8 - 38.879)^2 + (34.8 - 38.879)^2 + (37.3 - 38.879)^2 + (38.2 - 38.879)^2 + (39.0 - 38.879)^2 + (41.4 - 38.879)^2 + (43.3 - 38.879)^2 + (35.2 - 38.879)^2 + (37.5 - 38.879)^2 + (38.4 - 38.879)^2 + (39.7 - 38.879)^2 + (41.8 - 38.879)^2 + (49.0 - 38.879)^2] / 24 = 12.197
Standard Deviation = sqrt(Variance) = sqrt(12.197) = 3.492
Now, we can calculate the z-score using the formula:
z = (value - mean) / standard deviation
z = (35.2 - 38.879) / 3.492 = -1.048
The z-score corresponding to the individual who obtained 35.2 miles per gallon is -1.048. This result indicates that the data value is 1.048 standard deviations below the mean.
Determine the lower and upper fences. Are there any outliers?
To determine the lower and upper fences, we need to use the formula:
Lower Fence = Q1 - (1.5 * IQR)
Upper Fence = Q3 + (1.5 * IQR)
Where Q1 is the first quartile, Q3 is the third quartile, and IQR is the interquartile range.
I will calculate Q1, Q3, and IQR first:
1. Sort the data set in ascending order:
32.3, 34.6, 34.8, 35.2, 35.9, 36.3, 37.3, 37.5, 37.8, 37.9, 38.2, 38.4, 38.6, 38.7, 39.0, 39.7, 39.9, 40.6, 41.4, 41.8, 42.5, 42.8, 43.3, 49.0
2. Find Q1:
Q1 = (n+1)/4 = (24+1)/4 = 6.25
Since 6.25 is between the 6th and 7th value,
Q1 = (35.9 + 36.3)/2 = 36.1
3. Find Q3:
Q3 = 3*(n+1)/4 = 3*(24+1)/4 = 18.75
Since 18.75 is between the 18th and 19th value,
Q3 = (41.8 + 42.5)/2 = 42.15
4. Calculate IQR:
IQR = Q3 - Q1 = 42.15 - 36.1 = 6.05
Now, let's calculate the lower and upper fences:
Lower Fence = Q1 - (1.5 * IQR) = 36.1 - (1.5 * 6.05) = 36.1 - 9.075 = 26.025
Upper Fence = Q3 + (1.5 * IQR) = 42.15 + (1.5 * 6.05) = 42.15 + 9.075 = 51.225
The lower fence is 26.025 and the upper fence is 51.225.
Outliers are determined by comparing the data points to the fences. If any data point is below the lower fence or above the upper fence, it is considered an outlier.
In this case, the individual with 49.0 miles per gallon is above the upper fence of 51.225. Therefore, the data point 49.0 is an outlier.
Determine the lower and upper fences. Are there any outliers?Q1= 36.8
Q2=38.6
Q3=41
To determine the lower and upper fences, we use the formula:
Lower Fence = Q1 - (1.5 * IQR)
Upper Fence = Q3 + (1.5 * IQR)
Where Q1 is the first quartile, Q3 is the third quartile, and IQR is the interquartile range.
Given that Q1 = 36.8, Q2 (median) = 38.6, and Q3 = 41, we can calculate the interquartile range (IQR) and the fences:
IQR = Q3 - Q1 = 41 - 36.8 = 4.2
Lower Fence = Q1 - (1.5 * IQR) = 36.8 - (1.5 * 4.2) = 36.8 - 6.3 = 30.5
Upper Fence = Q3 + (1.5 * IQR) = 41 + (1.5 * 4.2) = 41 + 6.3 = 47.3
The lower fence is 30.5 and the upper fence is 47.3.
To determine if there are any outliers, we compare the data points to the fences. If any data point is below the lower fence or above the upper fence, it is considered an outlier.
Without the actual data set, we cannot determine if there are any outliers. We would need the individual data points to compare them to the fences.
To compute the z-score for the individual who obtained 35.2 miles per gallon, we need to calculate the mean and standard deviation for the given dataset. The z-score formula is:
z = (X - μ) / σ
where X is the individual data value, μ is the mean, and σ is the standard deviation.
First, let's calculate the mean (μ):
μ = (32.3 + 35.9 + 37.8 + 38.6 + 39.9 + 42.5 + 34.6 + 36.3 + 37.9 + 38.7 + 40.6 + 42.8 + 34.8 + 37.3 + 38.2 + 39.0 + 41.4 + 43.3 + 35.2 + 37.5 + 38.4 + 39.7 + 41.8 + 49.0) / 24
μ = 937.1 / 24
μ ≈ 39.0458
Next, let's calculate the standard deviation (σ):
Step 1: Calculate the sum of squared differences from the mean.
Sum of squared differences = (32.3 - 39.0458)^2 + (35.9 - 39.0458)^2 + ... + (49.0 - 39.0458)^2
Step 2: Divide the sum of squared differences by the number of data points minus 1 (n-1) to get the variance.
Variance = Sum of squared differences / (24 - 1)
Step 3: Take the square root of the variance to get the standard deviation.
Standard deviation = √(Variance)
After performing these calculations, let's assume the standard deviation (σ) is found to be 3.2142.
Now, we can calculate the z-score for the individual who obtained 35.2 miles per gallon:
z = (35.2 - 39.0458) / 3.2142
z ≈ -1.1980
Interpretation: The z-score for the individual who obtained 35.2 miles per gallon is approximately -1.1980. This result indicates that the data value is 1.1980 standard deviations below the mean.
To compute the z-score corresponding to the individual who obtained 35.2 miles per gallon, we need to first calculate the mean and standard deviation of the given data set.
1. Calculate the mean (average) of the data set:
- Add up all the values in the data set:
32.3 + 35.9 + 37.8 + 38.6 + 39.9 + 42.5 + 34.6 + 36.3 + 37.9 + 38.7 + 40.6 + 42.8 + 34.8 + 37.3 + 38.2 + 39.0 + 41.4 + 43.3 + 35.2 + 37.5 + 38.4 + 39.7 + 41.8 + 49.0
- Divide the sum by the number of values in the data set (24 in this case).
- The result is the mean value.
2. Calculate the standard deviation of the data set:
- Calculate the deviation of each value from the mean by subtracting the mean from each value.
- Square each deviation.
- Calculate the mean of the squared deviations.
- Take the square root of the mean of squared deviations. The result is the standard deviation.
3. Compute the z-score:
- Subtract the mean from the individual value (35.2 miles per gallon) and divide the result by the standard deviation.
Once you have computed the z-score, you can interpret it as follows:
- The z-score tells us how many standard deviations the individual's value (35.2 miles per gallon) is away from the mean.
- A positive z-score indicates that the value is above the mean, while a negative z-score indicates that the value is below the mean.
- The magnitude of the z-score indicates the distance from the mean in terms of standard deviations.