Suppose that a category of world class runners are known to run a marathon (26 m
ID: 3134943 • Letter: S
Question
Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 148 minutes with a standard deviation of 12 minutes. Consider 49 of the races. Let X = the average times of the 49 races. (a) Find the median of the average running times. ( That is, find the median of the sampling distribution of the average running times.) min (b) Find the slowest 10 percent of the average running times. ( Round the answer to the second decimal place.) min (c) Find the fastest 10 percent of the average running times. ( Round the answer to the second decimal place.) min
Explanation / Answer
a)
By central limit theorem, average running times are approximately normally distributed, so the mean is also the median.
Hence,
Median = 148 minutes [ANSWER]
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b)
Slowest 10% means the 90th percentile of time [more time is slower].
First, we get the z score from the given left tailed area. As
Left tailed area = 0.9
Then, using table or technology,
z = 1.281551566
As x = u + z * s / sqrt(n)
where
u = mean = 148
z = the critical z score = 1.281551566
s = standard deviation = 12
n = sample size = 49
Then
x = critical value = 150.1969455 [ANSWER]
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c)
First, we get the z score from the given left tailed area. As
Left tailed area = 0.1
Then, using table or technology,
z = -1.281551566
As x = u + z * s / sqrt(n)
where
u = mean = 148
z = the critical z score = -1.281551566
s = standard deviation = 12
n = sample size = 49
Then
x = critical value = 145.8030545 [ANSWER]
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