Question

Assume that your friend claims that he can run 20 km in less than an hour and a professional sports team will recruit him after graduation based on this outcome so he does not worry about his GPA at the university. You would like to test this claim and take a random sample of 34 runs this year and find that he averaged 58.2 minutes. Assume that the standard deviation of his runs is 8.5 minutes and it is normally distributed. a. State the null and alternative hypotheses. b. Test the hypotheses at the 5% significance level. What is the p-value of the test? c. What is your conclusion in the context of the question? d. Interpret the p-value you found above. e. What type of error you can commit here? Explain in the context of the question. Also, what are the implications of this error?

Answer 1

[Used R-Software]

# Assume that your friend claims that he can run 20 km in less than an hour
# mu< 60 (one hour=60 minutes)

# A random sample of 34 runs this year found that he averaged 58.2 minutes.
n=34 #sample size
xbar=58.2 #sample mean

# Assume that the standard deviation of his runs is 8.5 minutes and it is normally distributed.
s=8.5 #sample standard deviation

a. State the null and alternative hypotheses.
# Null hypothesis-H0: mu=60
# Alternative hypothesis: mu<60 (claim: Friend can run 20 km in less than an hour i.e. 60 minutes)

b. Test the hypotheses at the 5% significance level. What is the p-value of the test?
mu0=60
# Test statistic is given by

Z=(xbar-mu0)/(s/sqrt(n))
Z
# [1] -1.23479

pnorm(Z) #p-value
[1] 0.1084544

c. What is your conclusion in the context of the question?
The value of z is -1.23479. The value of p is 0.1084544.This p-value is greater than alpha(0.05), therefore we fail to Reject H0. The result is not significant at 5% level. Means, the claim of the friend is not true.

d. Interpret the p-value you found above.
p-value helps at arriving at conclusion. If p-value is less than alpha, H0 is rejected.

In other words,
High P values: your data are likely with a true null.
Low P values: your data are unlikely with a true null.
P values address only one question: how likely are your data, assuming a true null hypothesis? It does not measure support for the alternative hypothesis.