The randomly selected delivery times (in minutes) of two fast food restaurants are as follows.
Fast Food 1 Fast Food 2
42 40
38 37
37 40
39 41
38 38
40 46
45 41
41 37
40 40
Which of the following statements is false?
a Neither data set contains an outlier.
b Both fast food restaurants have the same mean delivery time of 40 minutes.
c Both fast food restaurants have exactly one modal delivery time of 40 minutes.
d Both fast food restaurants have the same median delivery time of 40 minutes.
c Both fast food restaurants have exactly one modal delivery time of 40 minutes. (Fast Food 1 has two modes, 38 and 40, while Fast Food 2 has one mode at 40)
To determine which statement is false, let's analyze each option:
a) Neither data set contains an outlier.
To identify outliers, we need to calculate the interquartile range (IQR) for each data set and then check for values beyond 1.5 times the IQR.
For Fast Food 1:
- Q1 (first quartile) = 38
- Q3 (third quartile) = 41
- IQR = Q3 - Q1 = 41 - 38 = 3
- Values beyond 1.5 times the IQR:
- No values are beyond 1.5 times the IQR for Fast Food 1.
For Fast Food 2:
- Q1 = 38
- Q3 = 41
- IQR = Q3 - Q1 = 41 - 38 = 3
- Values beyond 1.5 times the IQR:
- No values are beyond 1.5 times the IQR for Fast Food 2.
Since neither data set contains any values beyond 1.5 times the IQR, both data sets do not have outliers. Therefore, statement a) is true.
b) Both fast food restaurants have the same mean delivery time of 40 minutes.
To check if the mean delivery times are the same, we calculate the mean (average) for each data set.
For Fast Food 1:
- Mean = (42 + 38 + 37 + 39 + 38 + 40 + 45 + 41 + 40) / 9 = 360 / 9 = 40 minutes
For Fast Food 2:
- Mean = (40 + 37 + 40 + 41 + 38 + 46 + 41 + 37 + 40) / 9 = 360 / 9 = 40 minutes
Both data sets have the same mean delivery time of 40 minutes. Therefore, statement b) is true.
c) Both fast food restaurants have exactly one modal delivery time of 40 minutes.
Modal value refers to the value that appears most frequently in a data set.
For Fast Food 1: The number 40 appears twice and is the mode.
For Fast Food 2: The number 40 appears three times and is the mode.
Since the number of modal values is not the same for both data sets, statement c) is false.
d) Both fast food restaurants have the same median delivery time of 40 minutes.
To find the median, we need to arrange the values in ascending order and find the middle value.
For Fast Food 1: Sorted values - 37, 38, 38, 39, 40, 40, 41, 41, 45
- Median = 40 minutes
For Fast Food 2: Sorted values - 37, 37, 38, 40, 40, 40, 41, 41, 46
- Median = 40 minutes
Both data sets have the same median delivery time of 40 minutes. Therefore, statement d) is true.
Based on our analysis, the false statement is c) Both fast food restaurants have exactly one modal delivery time of 40 minutes.
To determine which statement is false, we need to examine each option and compare the data for Fast Food 1 and Fast Food 2.
a) Neither data set contains an outlier.
To verify this, we need to identify any values that are significantly different from the other values in the data set. Looking at the data, we do not see any values that are distinctly larger or smaller than the others. Therefore, statement a) is likely true.
b) Both fast food restaurants have the same mean delivery time of 40 minutes.
To check this, we need to calculate the mean (average) delivery time for both data sets. For Fast Food 1, the sum of the delivery times is 42 + 38 + 37 + 39 + 38 + 40 + 45 + 41 + 40 = 360. Dividing this sum by the number of values (9), we get a mean delivery time of 40 minutes. For Fast Food 2, the sum of the delivery times is 40 + 37 + 40 + 41 + 38 + 46 + 41 + 37 + 40 = 360. Again, dividing by 9 gives a mean delivery time of 40 minutes. Therefore, statement b) is true.
c) Both fast food restaurants have exactly one modal delivery time of 40 minutes.
The mode is the value that appears most frequently in the data set. In Fast Food 1, the modal delivery time is 38, as it appears twice. In Fast Food 2, the modal delivery time is 40, as it also appears twice. Therefore, statement c) is false.
d) Both fast food restaurants have the same median delivery time of 40 minutes.
The median is the middle value when the data set is ordered from smallest to largest. For Fast Food 1, when we arrange the data in ascending order, we get 37, 38, 38, 39, 40, 40, 41, 41, 45. The middle value is 40, so the median is 40 minutes. For Fast Food 2, when we arrange the data in ascending order, we get 37, 37, 38, 40, 40, 40, 41, 41, 46. Again, the middle value is 40, so the median is also 40 minutes. Therefore, statement d) is true.
After examining all the options, we find that the false statement is c) Both fast food restaurants have exactly one modal delivery time of 40 minutes.