At the Department of Education at UR University student records suggest that the

Question

At the Department of Education at UR University student records suggest that the population of students spends an average of 5.5 hours per week playing organized sports. The population’s standard deviation is 2.2 hours per week. Based on a sample of 121 students, Healthy Lifestyles Incorporated (HLI) would like to apply the central limit theorem to make various statements.

a. Calculate the probability that the sample mean will be between 5.3 and 5.7 hours.

b. How strange would it be to obtain a sample mean greater than 6.5 hours?

Explanation / Answer

The mean of a sampling distribution is the same as the population. Therefore =5.5

The standard deviation of a sampling distribution is found by /n. This gives 2.2/121 = 0.2

We can use the central limit theorem to say that the distribution is approximately normal with a mean of 5.5 and a standard deviation of 0.2

Using a graphing calculator we can find the probability that the sample mean will be between 5.3 and 5.7 with normalcdf(5.3,5.7,5.5,0.2)=0.6827

To do it by hand we would find the z-scores of 5.3 and 5.7 with the formula z=(x-mean)/std. dev.

This gives the z-scores as -1 and 1. Then we use a z-score chart to find the corresponding values, and subtract the one for -1 from the one for 1 to find the probability that the sample mean will be BETWEEN 5.3 and 5.7. The answer will be off from the graphing calculator way because of rounding errors. Note that we could have used the 68-95-99.7 rule to approximate the probability.

Having a sample mean greater than 6.5 hours would be extremely rare since it is 5 standard deviations away from the mean.

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