Question

A sprint duathlon consists of a 5 km run, a 20 km bike ride, followed by another 5 km run. The mean finish time of all participants in a recent large duathlon was 1.67 hours with a standard deviation of 0.25 hours. Suppose a random sample of 30 participants was taken and the mean finishing time was found to be 1.59 hours with a standard deviation of 0.30 hours.

(a) Identify the parameters in this situation.
(b) Identify the statistics in this situation.
(c) Suppose we were to make a histogram of the finishing times of all participants in the duathlon. Which of the following would the histogram be a display of?
i. Population distribution
iii. Distribution of a sample
ii. Sampling distribution of means
(d) Suppose the process of taking random samples of size 30 is repeated 200 times and a histogram of 200 sample means is created. Which of the following would the histogram be a display of?
i. Population distribution
ii. Distribution of a sample
iii. Sampling distribution of means
(e) What is the standard error for the mean finish time of 30 randomly selected participants?
(f) If one participant is selected at random, what is the probability that his/her time will be less than 1.59 hours.
(g) What is the probability that another sample of 30 randomly selected participants will have a mean finishing time of 1.59 hours or less?

Answer 1

A.) Parameter = 1.67 ; 0.25

B.) Statistic = 1.59, 0.30

C.) Population distribution

D.) Sampling distribution of means

E.) 0.058

F.) 0.395

G.) 0.072

Explanation:

A.) The parameters are measured quantities or characteristics derived from the population. The parameter here describes the mean and standard deviation of all participants ;

Population mean = 1.67

Population standard deviation = 0.25

B.) Statistic are measured quantities which are characteristics of the sample. Here, we have the sample mean and standard deviation

Sample mean = 1.59

Sample standard deviation = 0.30

C.) The population distribution is the plot of the histogram prepared using data from the population (all participants)

D.) The plot of the mean value obtained from the repeated measurement or simulation of the sample data is called sampling distribution of the means.

E.) Mean finish time of 30 randomly participants :

Sample size, n = 30

Standard Error = s/√n

Standard Error = 0.30 /√30 = 0.0548

F.) If one participant is selected at random, what is the probability that his/her time will be less than 1.59 hours

P(x < 1.59)

Z = (xbar - μ) / σ ; (1.59 - 1.67) / 0.30 = - 0.267

P(Z < - 0.267) = 0.395 (Z probability calculator)

G.) P(x < 1.59)

Z = (xbar - μ) / s ; (1.59 - 1.67) / (0.30/√30) = - 1.46

P(Z < -1.46) = 0.072 (Z probability calculator)