Question

Georgianna claims that in a small city renowned for its music school, the average child

takes less than 3 years of piano lessons. We have a random sample of 60 children from

the city, with a mean of 3.6 years of piano lessons and a standard deviation of 1.2

years.

1. Write the null and alternative hypothesis for this question

       b. Construct a 95% confidence interval for the number of years students

            in the city take piano lessons and interpret it in the context of the data.

      c. Does the null hypothesis proposed population mean fall within the 95%

          confidence interval?

      d. What is your conclusion?

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 3
Alternative Hypothesis, Ha: μ < 3

b)

sample mean, xbar = 3.6
sample standard deviation, s = 0.2
sample size, n = 60
degrees of freedom, df = n - 1 = 59

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.001

ME = tc * s/sqrt(n)
ME = 2.001 * 0.2/sqrt(60)
ME = 0.052

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (3.6 - 2.001 * 0.2/sqrt(60) , 3.6 + 2.001 * 0.2/sqrt(60))
CI = (3.5 , 3.7)

we are 95% confidnt that the number of years students

in the city take piano lessons are between 3.5 and 3.7

c)

It does not fall within the 95% confidence interval

d)

Reject the null hypothesis

There is sufficient evidence to conclude that the claim