Listed below are the duration's (in hours) of a simple random sample of all flights of a space shuttle program. Find the range, variance, and standard deviation for the sample data. Is the lowest duration time unusual? Why or why not?
71, 100, 237, 199, 164, 269, 193, 379, 252, 233, 388, 331, 223, 240, 0
I think I know how to do this but there are so many steps that I'm not sure what becomes before what. Please Help, this is my last class and then I graduate.
Didn't I answer this previously? Please only post your questions once. Repeating posts will not get a quicker response. In addition, it wastes our time looking over reposts that have already been answered in another post. Thank you.
Range = highest score minus lowest
Find the mean = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability in the smaller area related to the Z score.
I'll let you do the calculations.
Yes, and maybe I should have made myself clearer but my husband just killed himself and this is my last class and then I graduate and I never took Algebra or anything beyond basic math in high school. What or all I needed to know was the steps not the answers. Thank you
To find the range, variance, and standard deviation for the sample data, follow these steps:
1. Range: The range is the difference between the highest and lowest values in the data set.
- Highest value: 388
- Lowest value: 0
- Range = Highest value - Lowest value = 388 - 0 = 388
2. Variance: The variance measures how spread out the data is from the mean.
- Calculate the mean (average) of the data set:
- Sum of all durations = 71 + 100 + 237 + 199 + 164 + 269 + 193 + 379 + 252 + 233 + 388 + 331 + 223 + 240 + 0 = 3318
- Mean = Sum of all durations / Number of data points = 3318 / 15 = 221.2
- Calculate the squared difference between each duration and the mean:
- (71 - 221.2)^2 = 25281.44
- (100 - 221.2)^2 = 14786.24
- (237 - 221.2)^2 = 250.24
- ...
- (240 - 221.2)^2 = 352.84
- (0 - 221.2)^2 = 48944.24
- Calculate the sum of all squared differences: 25281.44 + 14786.24 + 250.24 + ... + 352.84 + 48944.24 = 354424.16
- Variance = Sum of squared differences / (Number of data points - 1) = 354424.16 / (15 - 1) = 23628.27 (rounded to two decimal places)
3. Standard Deviation: The standard deviation is the square root of the variance. It measures the average distance between each data point and the mean.
- Standard Deviation = √Variance = √23628.27 = 153.6 (rounded to one decimal place)
Now, let's determine if the lowest duration time (0 hours) is unusual.
To assess whether the lowest duration time is unusual, we can use the concept of z-scores. A z-score measures how many standard deviations a data point is from the mean.
- Calculate the z-score for the lowest duration time (0 hours):
- z-score = (Data point - Mean) / Standard Deviation = (0 - 221.2) / 153.6 = -1.427
Typically, z-scores greater than 3 or smaller than -3 are considered unusual or outliers. In this case, the z-score of -1.427 suggests that the lowest duration time is not unusually low.
To find the range, variance, and standard deviation for the given sample data, follow these steps:
1. Range:
The range is the difference between the highest and lowest values in the data set.
- First, sort the data from lowest to highest: 0, 71, 100, 164, 193, 199, 223, 233, 237, 240, 252, 269, 331, 379, 388.
- The lowest value is 0, and the highest value is 388.
- Subtract the lowest value from the highest value: 388 - 0 = 388.
- So, the range of the sample data is 388.
2. Variance:
Variance measures the dispersion or variability of the data.
- First, find the mean (average) of the data. Add up all the values and divide by the total count.
(0 + 71 + 100 + 164 + 193 + 199 + 223 + 233 + 237 + 240 + 252 + 269 + 331 + 379 + 388) / 15 = 2029 / 15 = 135.27 (rounded to two decimal places).
- Next, calculate the difference between each data point and the mean, square the differences, and sum them up.
(0 - 135.27)^2 + (71 - 135.27)^2 + (100 - 135.27)^2 + ... + (388 - 135.27)^2 = 669,160.82 (rounded to two decimal places).
- Divide the sum of squared differences by the total count of data points.
669,160.82 / 15 = 44,610.72 (rounded to two decimal places).
- Therefore, the variance of the sample data is 44,610.72.
3. Standard Deviation:
The standard deviation is the square root of the variance and provides a measure of the spread of the data.
- Take the square root of the variance calculated in the previous step: √44,610.72 ≈ 211.18 (rounded to two decimal places).
- The standard deviation of the sample data is approximately 211.18.
Now, to determine if the lowest duration time of 0 is unusual, we can compare it to the mean and standard deviation.
- The mean is 135.27, and the standard deviation is 211.18.
- The lowest duration time of 0 is considerably lower than the mean and falls far outside the range of one standard deviation.
- Although without further information it is challenging to definitively label it, a duration time of 0 could be considered unusual or potentially an outlier in this context.
Congratulations on your upcoming graduation!