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
Please help, I am so confused. Thank you
Find the mean first = 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 related to the Z score. How likely is that lowest score?
I'll let you do the calculations.
To find the range, variance, and standard deviation of the given sample data, follow these steps:
1. Range: The range is the difference between the largest and smallest values in the data set. To find it, you need to first sort the data in ascending order. The sorted data set is as follows:
0, 71, 100, 164, 193, 199, 223, 233, 237, 240, 252, 269, 331, 379, 388
The range is the difference between the largest value (388) and the smallest value (0), which is 388 - 0 = 388.
2. Variance: Variance measures the variability or spread of data points around the mean. To calculate it, you need to perform the following steps:
a. Calculate the mean (average) of the data set. Add up all the values and divide by the number of values (15 in this case).
Total sum = 0 + 71 + 100 + 164 + 193 + 199 + 223 + 233 + 237 + 240 + 252 + 269 + 331 + 379 + 388 = 3709
Mean = 3709 / 15 = 247.27 (rounded to two decimal places)
b. Calculate the squared difference between each value and the mean. Square the difference between each value and the mean.
Squared difference for each value:
(0 - 247.27)², (71 - 247.27)², (100 - 247.27)², ..., (388 - 247.27)²
c. Calculate the sum of all the squared differences from the previous step.
Sum of squared differences = (0 - 247.27)² + (71 - 247.27)² + (100 - 247.27)² + ... + (388 - 247.27)² = 4252070.91 (rounded to two decimal places)
d. Divide the sum of squared differences by the number of values (15) to get the variance.
Variance = Sum of squared differences / Number of values = 4252070.91 / 15 = 283471.39 (rounded to two decimal places)
3. Standard Deviation: The standard deviation is the square root of the variance. To calculate it, take the square root of the variance calculated in the previous step.
Standard Deviation = √Variance = √283471.39 = 532.35 (rounded to two decimal places)
Regarding whether the lowest duration time (0 hours) is unusual or not, it depends on the context and the specific criteria of what is considered unusual. However, it is worth noting that a flight duration of 0 hours is outside the range of the other data points and could be considered unusual or an outlier.