The accompanying data represent the miles per gallon of a random sample of cars with a three-cylinder, 1.0 liter engine.
(a)
Compute the z-score corresponding to the individual who obtained 36.3
miles per gallon. Interpret this result.
(b)
Determine the quartiles.
(c)
Compute and interpret the interquartile range, IQR.
(d)
Determine the lower and upper fences. Are there any outliers?
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32.6
35.8
38.0
38.6
39.8
42.3
34.6
36.3
38.1
39.0
40.5
42.7
34.7
37.4
38.3
39.2
41.5
43.7
35.6
37.6
38.5
39.5
41.7
48.9
(a) To compute the z-score corresponding to an individual who obtained 36.3 miles per gallon, we need to calculate the mean and standard deviation of the sample.
The mean (µ) can be found by summing all the values and dividing by the total number of observations:
µ = (32.6 + 35.8 + 38.0 + 38.6 + 39.8 + 42.3 + 34.6 + 36.3 + 38.1 + 39.0 + 40.5 + 42.7 + 34.7 + 37.4 + 38.3 + 39.2 + 41.5 + 43.7 + 35.6 + 37.6 + 38.5 + 39.5 + 41.7 + 48.9) / 24 = 38.2625
The standard deviation (σ) can be found using the formula:
σ = √(1/(n-1) * Σ(xi - µ)²)
where xi represents each value in the sample, µ is the mean, and n is the number of observations.
σ = √(1/(24-1) * [(32.6-38.2625)² + (35.8-38.2625)² + (38.0-38.2625)² + ... + (41.7-38.2625)² + (48.9-38.2625)²])
= √(1/23 * [(-5.6625)² + (-2.4625)² + (-0.2625)² + ... + (3.4375)² + (10.6375)²])
= √(1/23 * [32.044 + 6.066 + 0.069 + ... + 11.824 + 112.884])
= √(1/23 * 263.934)
≈ 3.006
Now, we can calculate the z-score using the formula:
z = (x - µ) / σ
For x = 36.3:
z = (36.3 - 38.2625) / 3.006 ≈ -0.654
So, the z-score for 36.3 miles per gallon is approximately -0.654.
Interpretation: The z-score tells us how many standard deviations the value of 36.3 is from the mean of the sample. In this case, a z-score of -0.654 means that 36.3 miles per gallon is about 0.654 standard deviations below the mean.
(b) To determine the quartiles, we need to sort the data in ascending order first:
32.6, 34.6, 34.7, 35.6, 35.8, 36.3, 37.4, 37.6, 38.0, 38.1, 38.3, 38.5, 38.6, 39.0, 39.2, 39.5, 39.8, 40.5, 41.5, 41.7, 42.3, 42.7, 43.7, 48.9
The quartiles divide the data into four equal parts.
First Quartile (Q1) is the median of the lower half of the data: 36.3, 37.4, 34.7, 38.0, 35.8, 38.1, 34.6, 38.3, and 37.6. The median of this lower half is 36.8.
Third Quartile (Q3) is the median of the upper half of the data: 39.0, 39.2, 39.5, 39.8, 40.5, 41.5, 41.7, 42.3, 42.7, 43.7, and 48.9. The median of this upper half is 41.7.
So, the quartiles are:
Q1 ≈ 36.8
Q3 ≈ 41.7
(c) The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 - Q1.
IQR ≈ 41.7 - 36.8 = 4.9
Interpretation: The interquartile range of approximately 4.9 indicates that the middle 50% of the data, between the first and third quartiles, has a spread of 4.9 miles per gallon.
(d) To determine the lower and upper fences, we can use the formula:
Lower Fence = Q1 - 1.5 * IQR
Upper Fence = Q3 + 1.5 * IQR
Lower Fence ≈ 36.8 - 1.5 * 4.9 ≈ 29.65
Upper Fence ≈ 41.7 + 1.5 * 4.9 ≈ 48.65
Since there is no observation below the lower fence or above the upper fence, there are no outliers within the given data.
If a data value is larger than the mean, the z-score is positive. If a data value is smaller than the mean, the z-score is negative. If the data value equals the mean, the z-score is zero. A z-score measures the number of standard deviations an observation is above or below the mean. Remember to type integers or decimals rounded to two decimal places.
Yes, you are correct. If a data value is larger than the mean, the z-score is positive. If a data value is smaller than the mean, the z-score is negative. If the data value equals the mean, the z-score is zero. The z-score measures the number of standard deviations an observation is above or below the mean.
(b) Determine the quartiles.
To determine the quartiles, we need to sort the data in ascending order first:
32.6, 34.6, 34.7, 35.6, 35.8, 36.3, 37.4, 37.6, 38.0, 38.1, 38.3, 38.5, 38.6, 39.0, 39.2, 39.5, 39.8, 40.5, 41.5, 41.7, 42.3, 42.7, 43.7, 48.9
There are 24 observations in total.
To find the first quartile (Q1), we need to find the median of the lower half of the data. Since there is an even number of observations, the lower half consists of the first 12 observations. The median will then be the average of the 6th and 7th values:
Q1 = (35.8 + 36.3) / 2 = 36.05
To find the third quartile (Q3), we need to find the median of the upper half of the data. Again, since there is an even number of observations, the upper half consists of the last 12 observations. The median will be the average of the 6th and 7th values:
Q3 = (40.5 + 41.5) / 2 = 41.00
Therefore, the quartiles for the given data are:
Q1 = 36.05
Q3 = 41.00
What about Q2
Apologies for the oversight. Q2 represents the median of the entire dataset, also known as the second quartile.
In our sorted dataset, we have 24 observations. Since this is an even number, we will find the average of the two middle values to calculate Q2.
The two middle values are the 12th and 13th observations:
Q2 = (38.1 + 38.3) / 2 = 38.2
So, the second quartile (Q2) for the given data is 38.2.
I got
Q1= 36.85
Q2=38.55
Q3=41
Apologies for the confusion. Thank you for providing the correct calculations.
Using the given data, the quartiles are indeed:
Q1 = 36.85
Q2 = 38.55
Q3 = 41.00
Thank you for catching the error, and I apologize for any confusion caused.
(c) Compute and interpret the interquartile range, IQR. Select the correct choice below and fill in the answer box to complete your choice.
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 - Q1.
Using the correct values for Q1 and Q3 (Q1 = 36.85, Q3 = 41.00), we can calculate the IQR as follows:
IQR = Q3 - Q1
= 41.00 - 36.85
= 4.15
Therefore, the interquartile range (IQR) is approximately 4.15.
(d) Determine the lower and upper fences. Are there any outliers?
To determine the lower and upper fences, we can use the following formulas:
Lower Fence = Q1 - 1.5 * IQR
Upper Fence = Q3 + 1.5 * IQR
Using the correct values for Q1 (36.85), Q3 (41.00), and the calculated value for IQR (4.15), we can calculate the lower and upper fences as follows:
Lower Fence = 36.85 - 1.5 * 4.15
≈ 30.68
Upper Fence = 41.00 + 1.5 * 4.15
≈ 47.53
To identify outliers, we compare each data point to the lower and upper fences. If a data point is below the lower fence or above the upper fence, it is considered an outlier.
Looking at the dataset, we can see that there is one data point that exceeds the upper fence (48.9). Therefore, 48.9 is considered an outlier in this dataset.
In summary:
- The lower fence is approximately 30.68.
- The upper fence is approximately 47.53.
- There is one outlier in the dataset (48.9).