A psychologist believes that if classical music is played du

A psychologist believes that if classical music is played during a test, test scores will increase. In the past, the mean test score was 60. A random sample of 49 individuals wrote the test while classical music was in the background. The sample results showed a mean test score of 62 and a standard deviation of 6.3. Using a 5% level of significance, can the psychologist conclude that the mean test score has increased with classical music? Formulate and test the appropriate hypotheses. Use the critical value approach.

Solution

let X be the random variable denoting the scores of a test while classical music was in the background.

the assumption is that X follows a normal distribution with mean=E[X]=mu (say) and variance=V[X]=sigma²

where mu and sigma are unknown.

A psychologist believes that if classical music is played during a test, test scores will increase. In the past, the mean test score was 60.

so basically we are to test whether mu is greater than 60 or not.

so the null hypothesis is H0:mu=60 vs the alternative hypothesis is H1: mu>60

to test the hypothesis a random sample of size=n=49 is taken.

the sample mean is Xbar=62 and the sample standard deviation is s=6.3

since the population standard deviation sigma is unknown,

the test statistic for testing H0 vs H1 is

T=sqrt(n)*(Xbar-60)/s which under H0 follows a t distribution with df=n-1=48

now by critical value approach using a 5% level of significance and as the alternative hypothesis is a right sided hypothesis H0 is rejected iff

t>t_alpha;n-1   where t is the observed value of T and t_alpha;n-1 is the upper 100*(1-alpha)% point[critical value] of a t distribution with df n-1

now we have n=49, Xbar=62 s=6.3 alpha=0.05

so t=sqrt(49)*(62-60)/6.3=2.222

and t_0.05;48=1.67722

so we have t>t_0.05;48

hence H0 is rejected and the conlcusion is that mu>60

hence at 5% level of significance based on the given data at hand the psychologist should conclude that the mean test score has increased with classical music. [answer]