Question

g Georgianna claims that in a small city renowned for its music school, the average child takes at least 5 years of piano lessons. We have a random sample of 20 children from the city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years. What is the 90% confidence interval for the average number of years students take piano lessons in this city? Group of answer choices

Answer 1

Answer:

[3.75 years, 5.45 years]

Step-by-step explanation:

The 90% confidence interval is given by the interval

$\large [\bar x-t^*\frac{s}{\sqrt n}, \bar x+t^*\frac{s}{\sqrt n}]$

where

$\large \bar x$ is the sample mean  

s is the sample standard deviation  

n is the sample size

$\large t^*$ is the 0.05 (5%) both-sided (**) critical value for the Student's t-distribution with 19 degrees of freedom (sample size -1), which is an approximation to the Normal distribution for small samples (n≤ 30).

Either by using a table or the computer, we find  

$\large t^*= 1.729$

and our 90% confidence interval is

$\large [4.6-1.729*\frac{2.2}{\sqrt{20}}, 4.6+1.729*\frac{2.2}{\sqrt{20}}]=\boxed{[3.75,5.45]}$

(**) 5% of the area to the left of $\large -t^*$ and 5% to right of $\large t^*$ for a total of 90% inside the interval $\large [-t^*,+t^*]$