- A researcher is investigating the accuracy of IQ tests. One company offers IQ tests that it claims will give any individual's IQ with a standard deviation of 5
The researcher takes these tests 9 times with the following results
$$123 , \quad 118 , \quad 127 , \quad 120 , \quad 134 , \quad 120 , \quad 118 , \quad 135 , \quad 121$$
- Find the sample mean, \(\bar { x }\), and the sample variance, \(s ^ { 2 }\), of these scores.
(2)
Given that any individual's IQ scores on these tests are independent and have a normal distribution, - use the hypotheses
$$\mathrm { H } _ { 0 } : \sigma ^ { 2 } = 25 \text { against } \mathrm { H } _ { 1 } : \sigma ^ { 2 } > 25$$
to test the company's claim at the \(5 \%\) significance level.
(4)
Gurdip works for the company and has taken these IQ tests 12 times. Gurdip claims that the sample variance of these 12 scores is \(s ^ { 2 } = 8.17\) - Use this value of \(s ^ { 2 }\) to calculate a \(95 \%\) confidence interval for the variance of Gurdip's IQ test scores.
[0pt]
[You may use \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 3.816 \right) = 0.975\) and \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 21.920 \right) = 0.025\) ] - Assuming that \(\sigma ^ { 2 } = 25\), comment on Gurdip's claim.