Answer :
The z-score for the data value 38.1 is 1.34, which indicates that the data value is 1.34 standard deviations above the mean. The quartiles are 30.9, 35.7, 38.7. The interquartile range (IQR) is 7.8. This indicates that the middle 50% of the data falls between 30.9 and 38.7 miles per gallon. The lower and upper fences are 18.9 and 56.7 respectively. There are no outliers in this data set as all the data values lie within the lower and upper fences.
(a) Compute the z-score corresponding to the individual who obtained 38.1 miles per gallon.
The given data represent the miles per gallon of a random sample of cars with a three-cylinder, 1.0-liter engine. We need to calculate the z-score for the data value 38.1.
The formula for calculating the z-score is: z = (x - μ) / σ,
where x is the data value, μ is the mean of the sample, and σ is the standard deviation of the sample.
Mean = (29.4 + 30.0 + 30.6 + 31.2 + 31.8 + 32.4 + 33.0 + 33.6 + 34.2 + 34.8 + 35.4 + 36.0 + 36.6 + 37.2 + 37.8 + 38.4 + 39.0 + 39.6 + 40.2 + 40.8) / 20
Mean = 34.12 (rounded to two decimal places)
Standard deviation:σ = √[∑(x - μ)² / (n - 1)]
σ = √[834.166 / 19]
σ = 2.97 (rounded to two decimal places)
z-score = (38.1 - 34.12) / 2.97
z-score = 1.34 (rounded to two decimal places)
The z-score for the data value 38.1 is 1.34, which indicates that the data value is 1.34 standard deviations above the mean.
(b) Determine the quartiles.
Quartiles divide the data set into four equal parts. There are three quartiles, Q1, Q2, and Q3. Q2 is the median of the data set. Q1 is the median of the lower half of the data set, and Q3 is the median of the upper half of the data set.
To determine the quartiles for the given data set, we need to arrange the data values in order.
29.4, 30, 30.6, 31.2, 31.8, 32.4, 33, 33.6, 34.2, 34.8, 35.4, 36, 36.6, 37.2, 37.8, 38.4, 39, 39.6, 40.2, 40.8
n = 20
Q2 = median
Q2 = (35.4 + 36) / 2
Q2 = 35.7 (rounded to one decimal place)
Q1 = median of the lower half
Q1 = (30.6 + 31.2) / 2
Q1 = 30.9 (rounded to one decimal place)
Q3 = median of the upper half
Q3 = (38.4 + 39) / 2
Q3 = 38.7 (rounded to one decimal place)
(c) Compute and interpret the interquartile range, IQR.
The interquartile range (IQR) is the range of the middle 50% of the data and is computed by subtracting the first quartile from the third quartile.
IQR = Q3 - Q1
IQR = 38.7 - 30.9
IQR = 7.8
The interquartile range (IQR) is 7.8. This indicates that the middle 50% of the data falls between 30.9 and 38.7 miles per gallon.
(d) Determine the lower and upper fences. The lower fence is calculated as Q1 - 1.5(IQR) = 30.9 - 1.5(7.8) = 18.9The upper fence is calculated as Q3 + 1.5(IQR) = 38.7 + 1.5(7.8) = 56.7
Outliers-
Any data value that is less than the lower fence or greater than the upper fence is considered an outlier. There are no outliers in this data set as all the data values lie within the lower and upper fences.
Learn more about the z-score from the given link-
https://brainly.com/question/30892911
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