Question

1. True or False: When using data from the same sample, the 95% confidence interval for µ will always support the results from a 2-sided, 1 sample t-test. Explain your reasoning.

Answer 1

Answer)

True

As in a 2 sided test, critical values are same used for confidence interval

And rejection region is on two tails same as confidence interval

For example

We have a two sided hypothesis

Ho : u = 5

Ha : u not equal to 5

We are given with the population s.d as 1

and sample size of 9 resulted in a mean of 6

As the population s.d is mentioned we can use standard normal z table to conduct the test and construct the interval as well

Test statistics z = (sample mean - claimed mean)/(s.d/√n)

Z = (6 - 5)/(1/√9) = 3

Suppose confidence level is 95%

So alpha = 1 - 0.95 = 0.05

From z table, P(z<-1.96) = P(z>1.96) = 0.025 (0.05/2)

Rejection region is reject Ho if test statistics is less than -1.96 or greater than 1.96

So reject Ho as 3 > 1.96

Now we will test it using confidence level

Critical value z is 1.96

Margin of error (MOE) = Z*S.D/√N = 1.96*1/√9 = 0.65

Interval is given by

(Mean - moe, mean + moe)

(5.35, 6.65)

As interval do not contain the null hypothesised value 5, we reject the null hypothesis Ho