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.
(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
5 answers
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.
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
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.
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.