Calculating Mid-Interval Values
- The mid-interval value for the class 70 to less than 80 is calculated as (70 + 80) / 2 = 75.
- Other mid-interval values are similarly calculated:
- 80 to <85: (80 + 85) / 2 = 82.5
- 85 to <90: 87.5
- 90 to <95: 92.5
- 95 to <105: 100
Estimating the Mean Running Time
- Use the mid-interval values as representative values for each class.
- Multiply each mid-interval value by its corresponding frequency:
- 75 * 11 = 825
- 82.5 * 51 = 4207.5
- 87.5 * 68 = 5950
- 92.5 * 47 = 4347.5
- 100 * 23 = 2300
- Sum of frequencies = 200
- Sum of products = 825 + 4207.5 + 5950 + 4347.5 + 2300 = 17630
- Estimated mean (x̄) = Total sum of products / Total frequency = 17630 / 200 = 88.15 minutes
Using a Calculator for One-Variable Statistics
- Input mid-interval values as list X.
- Input frequencies as list FR.
- Use one-variable statistics function with frequency list to compute mean.
- Result confirms mean running time is approximately 88.15 minutes.
Estimating the Interquartile Range (IQR) from Cumulative Frequency Graph
- Total movies = 200
- Q3 (75th percentile) position = 0.75 * 200 = 150th value
- Q1 (25th percentile) position = 0.25 * 200 = 50th value
Finding Q3
- From the graph, Q3 corresponds to a running time of approximately 91.5 minutes.
- This is calculated by interpolating between 90 and 95 minutes using small units of 0.5 minutes.
Finding Q1
- Q1 corresponds to approximately 84 minutes from the graph.
Calculating IQR
- IQR = Q3 - Q1 = 91.5 - 84 = 7.5 minutes
Determining if "Starfield" is an Outlier
- Starfield's running time = 100 minutes
- Calculate upper outlier boundary:
- Upper boundary = Q3 + 1.5 * IQR = 91.5 + 1.5 * 7.5 = 102.75 minutes
- Since 100 < 102.75, Starfield is not an outlier.
Summary
- The estimated mean running time of the 200 family movies is 88.15 minutes.
- The interquartile range is 7.5 minutes, indicating the middle 50% of running times fall within this range.
- Starfield's running time of 100 minutes is within the expected range and not considered an outlier.
For further insights into statistical concepts, you may find these resources helpful:
The running time t of 200 family movies are recorded in the following table. Write down the mid interval value of t
greater than or equal to 70 less than 80. Mid interval value equals 70 + 80 over
2. So we get 75. Calculate an estimate of the mean running
time of the 200 movies. First of all, we need to figure out mid interval value.
75 80 + 85 over 2 82.5 87.5
92.5 100 Then we need to put this into calculator to figure out the mean. Since
uh this is a frequency so only have one variable we need to go to one variable stats click
on one then no we will go to four name column one as a
P then column B as the frequency. So, FR. After you put in all the
values, click on an empty cell menu for statistics, one stats
calculations. This is a one variable stats because the second column is a
frequency. For one variable stats, number of leads is always one. Then enter. Right arrow button to get t as x1
list. for frequency list it's
a fr then enter so we get this xbar which is mean
is 88.15 since this is two marks so we need to show the work xbar equals
75 * 11 82.5 * 51 87.5
* 68 92.5 *
47 100 * 23 all over frequency add all these together equals 200
100 equals 88.15. This table is used to create the
following cumulative frequency graph. Use the cumulative frequency curve to estimate
the inter quartile range. Intercortile range IQR equals Q3 minus Q1.
Q3 it's 75%. So 75%
times cumulative frequency total is 200 equals
150. Get 150. Then going down. Then you need to
check the unit for running time from 70 to 75.
We count how many small units here. 75 minus 70 over 10 equals
0.5. So 90 plus we have three units of
0.5. 90 + 3 * 0.5 is 91.5. For Q1, the first quartile equals 25%. So
25% times the total number 200 equals
50. So get this 50. Draw this horizontal line. Then going
down 80 plus all together is a eight each is a
0.5. So we get 84 equals 91.5 minus
84 equals 7.5. Starfield is a movie in the data set and its running time is 100 minutes. Use
your answer to part B to estimate whether Starfield's running time is an outlier for this data. Justify your
answer. We have a lower outlier. We will use the
closest quartile minus
1.5 IQR higher
outlier closest quartile is Q3 add 1.5
IQR we will find the higher outline liar equals Q3 is
91.5 + 1.5 * 7.5 = 102.75.
Since 100 less than 102.75, it's a not an outlier.
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