Question

A simple random sample of men who regularly work out at Mitch's Gym is obtained and their resting pulse rates (in beats per minute) are listed below. 54 60 67 84 74 64 69 70 66 80 59 71 76 63 Use a 0.05 significance level to test the claim that the mean resting pulse rates is less than 72 beats per minute. Assume that the resting pulse rates of all men who work out at Mitch's Gym are normally distributed with standard deviation of 6.8 beats per minute.

(a) Identify the null hypothesis and alternative hypothesis.

(b) Find the test statistic.

(c) Calculate the P-value.

(d) Make conclusion about the null hypothesis and give the final conclusion that addresses the original claim.

Answer 1

Let the resting pulse rate be X

The data given on pulse rate is  54 ,60 ,67 ,84 ,74, 64, 69, 70, 66, 80, 59, 71, 76, 63

n=14(sample size)

Level of significance =0.05

It is also given that

a) Let be the mean resting pulse rate

NULL Hypothesis is

Alternative Hypothesis is  

b) We estimate

Since Xi are random sample hence Xi  follow , therefore, we know that which can be further normalised by subtracting mean and dividing by sd

Therefore the test statistics(Under H0)  will be Z=

c) Calculating p value

  68.357 (Calculated from data)

Test statistics for our sample (under Ho is true) is z =

Since, we have Z as our test statistics and z(small letter) is calculated value of test statistics and our test is lower tail test , hence

p value =P(Z<z|H0 is true)

p value =P(Z<z) =P(Z< -2.004)=0.023 (Here -Z~N(0,1) and z(small letter) calculated test statistics)

p value =0.023

d) Since the p value(0.023) is less than the level of significance (0.05 ) ,hence there is enough evidence  reject the Null Hyppthesis of and can say that mean resting pulse level is less than 72

So the final inference is that we can say with 95 % confidence that mean resting pusle rate is less than 72