Question

A study shows that the amount of time spent by millennials playing video games is 22.6 hours a month, with a standard deviation of 6.1 hours. A bright UWI stats student has doubts about the study’s results. She believes that they actually spend more time. The student tries to resolve her doubts, and collects a random sample of 60 millennials, asking them to keep a daily log of their video game playing habits. Millennials in the sample played an average of 24.2 hours per month. (a) If the null hypothesis is true, describe the sampling distribution of the mean number of hours spent playing video games. [5 marks] (b) Calculate the probability of randomly choosing a sample in which the average number of hours of video games played was 24.2 or more. [5 marks] (c) No hard and fast rule exists which divides the boundary between p-values for which we reject the null and those for which we feel the null is plausible. However p = 0.05 and p = Xi ~ Poisson : e?(2? ) (2?) X X ! , X = 0,1,2,… X 2 0.01 are two commonly used thresholds. Under these thresholds, should the student reject the null hypothesis? [5 marks] (d) Suppose the student doubted the study’s findings but had no prior expectation of whether they were too high or too low. Perhaps she should determine the probability of randomly choosing a sample in which the average number of hours spent playing video games was as extreme or more extreme that 24.2 hours. Should she reject the null hypothesis in this case? [5 marks] (e) Would a larger sample with the same mean of 24.2 have provided stronger evidence of a difference from the original study’s mean? Explain. [5 marks]

Answer 1

Given

Mean=22.6 SD=6.1

H0 : true mean =22.6

H1:true Mean >22.6

a)

As H0: mean =22.6

if H0 is true then mean is 22.6

so Sampling distribution of mean number of millenials playing video game is

with Mean =22.6 and SD =6.1 /Sqrt(60 ) =6.1/7.75 =0.79

b)

we have to find P(sample mean>24.2)=?

i.e

=P(Z>2.025)

=1-P(Z<2.025)

=1-0.9786=0.0214

c)

our Z statistic Value is

P value =P(Z>2.025) =0.0214

Since for Threshold Value =0.05

P value <0.05 Hence for Threshold Value 0.05 we reject the null Hypothesis

While for Threshold value =0.01 as P value >0.01 Hence we fail to reject the null hypothesis

d)

Here again as we calculated in (b) and (c) part

we calculate P-Value=0.0214 (as in C Part)

at level of significance 0.05 we reject the null hypothesis

e)

Yes

as

so

as n increase then Z will increase and we will get Smaller value of P-Value so there is more chance of rejection of Null hypithesis