The distribution of the time taken to run 1000 metres by users of a treadmill in a sports centre has mean 7.2 minutes and standard deviation
2.045 minutes.
1 Choose the option that is closest to the probability that the mean time taken to run 1000 metres for a sample of 30 users will be more than 8 minutes.
Options for Question 1
A 0.0160 B 0.2000 C 0.2486 D 0.8003 E 0.8030 F 0.8106
2 Choose the option that gives an approximate range of values,in minutes, symmetric about the mean, within which the meantime taken to run 1000 metres for approximately 95% of samples of 30 users will lie.
Options for Question 2
A (6.584, 7.816) B (6.469, 7.931) C (3.192, 11.208) D (6.640, 7.760) E (7.066, 7.334) F (4.088, 11.312)
To solve these questions, we can use the concept of the normal distribution and the properties of the z-score.
1. Probability calculation:
To find the probability that the mean time taken to run 1000 meters for a sample of 30 users will be more than 8 minutes, we need to calculate the z-score and then use the standard normal distribution table.
The formula for the z-score is:
z = (x - μ) / (σ / √n)
Where:
x = 8 (the desired mean time)
μ = 7.2 (the population mean time)
σ = 2.045 (the population standard deviation)
n = 30 (sample size)
The z-score is:
z = (8 - 7.2) / (2.045 / √30) = 2.2668
Now, we can use the standard normal distribution table (also known as the z-table) to find the probability associated with z = 2.2668. Looking up this z-value in the z-table, the closest probability is 0.0117.
Among the given answer choices, the option closest to 0.0117 is option A: 0.0160.
2. Range calculation:
To determine the range of values within which the mean time taken to run 1000 meters for approximately 95% of samples of 30 users will lie, we need to calculate the margin of error.
The margin of error is calculated by multiplying the critical value (obtained from the z-table for a desired confidence level) by the standard deviation of the sample mean.
For a 95% confidence level, we need to find the critical value associated with a central area of 0.95 in the standard normal distribution. Considering this is a two-tailed test, we will divide the remaining 0.05 by 2 to find the individual tail area of 0.025.
Looking up the z-table for a tail area of 0.025, the critical value is approximately 1.96.
The margin of error is:
ME = 1.96 * (2.045 / √30) = 0.7922
To find the range, we subtract and add the margin of error from the population mean:
Range = (7.2 - 0.7922, 7.2 + 0.7922) = (6.4078, 7.9922)
Among the given answer choices, the option that gives an approximate range of values symmetric about the mean, within which the mean time taken to run 1000 meters for approximately 95% of samples of 30 users will lie is option B: (6.469, 7.931).