Answer :
Sure! Here is how you can understand the results step by step:
### (a) Compute the [tex]\( z \)[/tex]-score for 39.3 miles per gallon
The [tex]\( z \)[/tex]-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value for which you're finding the [tex]\( z \)[/tex]-score, [tex]\( \mu \)[/tex] is the mean of the data, and [tex]\( \sigma \)[/tex] is the standard deviation.
1. The value of [tex]\( X \)[/tex] is 39.3.
2. The mean ([tex]\( \mu \)[/tex]) of the data set is approximately 39.1717.
3. The standard deviation ([tex]\( \sigma \)[/tex]) of the data is approximately 4.9707.
Plug these values into the formula:
[tex]\[ z = \frac{39.3 - 39.1717}{4.9707} \approx 0.10 \][/tex]
The [tex]\( z \)[/tex]-score of 0.10 indicates that the individual's miles per gallon value is approximately 0.10 standard deviations above the mean.
### (b) Determine the quartiles
- Q1 (First Quartile): 37.15
- Q2 (Second Quartile, or Median): 38.55
- Q3 (Third Quartile): 40.8
### (c) Compute and interpret the interquartile range (IQR)
The Interquartile Range (IQR) is the difference between the third and first quartile:
[tex]\[ \text{IQR} = Q3 - Q1 = 40.8 - 37.15 = 3.65 \][/tex]
The IQR represents the range within which the middle 50% of your data values lie.
### (d) Determine the lower and upper fences and identify any outliers
Lower and upper fences are calculated to detect outliers:
- Lower Fence: [tex]\( Q1 - 1.5 \times \text{IQR} = 37.15 - 1.5 \times 3.65 = 31.675 \)[/tex]
- Upper Fence: [tex]\( Q3 + 1.5 \times \text{IQR} = 40.8 + 1.5 \times 3.65 = 46.275 \)[/tex]
Any data values below the lower fence or above the upper fence are considered outliers. In this case:
- The outliers are any values greater than 46.275 or less than 31.675.
- The only value that falls outside this range is 49.1, which is an outlier.
These steps help us to understand the distribution and spread of the miles per gallon data, and to identify any anomalies.
### (a) Compute the [tex]\( z \)[/tex]-score for 39.3 miles per gallon
The [tex]\( z \)[/tex]-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value for which you're finding the [tex]\( z \)[/tex]-score, [tex]\( \mu \)[/tex] is the mean of the data, and [tex]\( \sigma \)[/tex] is the standard deviation.
1. The value of [tex]\( X \)[/tex] is 39.3.
2. The mean ([tex]\( \mu \)[/tex]) of the data set is approximately 39.1717.
3. The standard deviation ([tex]\( \sigma \)[/tex]) of the data is approximately 4.9707.
Plug these values into the formula:
[tex]\[ z = \frac{39.3 - 39.1717}{4.9707} \approx 0.10 \][/tex]
The [tex]\( z \)[/tex]-score of 0.10 indicates that the individual's miles per gallon value is approximately 0.10 standard deviations above the mean.
### (b) Determine the quartiles
- Q1 (First Quartile): 37.15
- Q2 (Second Quartile, or Median): 38.55
- Q3 (Third Quartile): 40.8
### (c) Compute and interpret the interquartile range (IQR)
The Interquartile Range (IQR) is the difference between the third and first quartile:
[tex]\[ \text{IQR} = Q3 - Q1 = 40.8 - 37.15 = 3.65 \][/tex]
The IQR represents the range within which the middle 50% of your data values lie.
### (d) Determine the lower and upper fences and identify any outliers
Lower and upper fences are calculated to detect outliers:
- Lower Fence: [tex]\( Q1 - 1.5 \times \text{IQR} = 37.15 - 1.5 \times 3.65 = 31.675 \)[/tex]
- Upper Fence: [tex]\( Q3 + 1.5 \times \text{IQR} = 40.8 + 1.5 \times 3.65 = 46.275 \)[/tex]
Any data values below the lower fence or above the upper fence are considered outliers. In this case:
- The outliers are any values greater than 46.275 or less than 31.675.
- The only value that falls outside this range is 49.1, which is an outlier.
These steps help us to understand the distribution and spread of the miles per gallon data, and to identify any anomalies.