Question

A company wants to determine whether its consumer product ratings ​(0−​10) have changed from last year to this year. The table below shows the​ company's product ratings from eight consumers for last year and this year. At α=​0.05, is there enough evidence to conclude that the ratings have​ changed? Assume the samples are random and​ dependent, and the population is normally distributed.

Rating (last year) 66 77 44 44 66 55 66 88

Rating​ (this year) 22 88 22 22 33 11 33 88

1) Calculate sd. (Round to three decimals as needed)

2) Calculate the test statistic (Round to three decimals as needed)

3)Decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim

Answer 1

Answer)

Ho : u1-u2 = 0

Ha : u1-u2 not equals 0

1)

First we need to find the mean and s.d for last year and this year

Last year

u1 = 63.25

s1 = 15.276

This year

u2 = 39.875

S2 = 30.517

N1 = N2 = 8

2)

Test statistics = (u1-u2)/standard error

Standard error = √{(s1^2/n1)+(s2^2/n2)}

After substitution

Test statistics is = 1.937

3)

As the population s.d is unknown, we will use t distribution to estimate the P-value

Degrees of freedom is = smaller of n1-1, n2-1

= 7

For df 7 and 1.937 test statistics

P-value is = 0.093949

As p-value is greater than the given significance level 0.05

We fail to reject the null hypothesis

So, we do not have enough evidence to support the claim that the ratings have changed