Question

NEED THIS ANSWERED AS SOON AS POSSIBLE

Use the data to calculate the sample variance, s. (Round your answer to five decimal places.) n = 7: 1.2, 3.4, 1.5, 2.1, 3.4, 2.7, 2.7 S2 Construct a 95% confidence interval for the population variance, 02. (Round your answers to two decimal places.) x to Test Ho: o2 = 0.8 versus H : 02 +0.8 using a = 0.05. State the test statistic. (Round your answer to two decimal places.) x2 = State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to two decimal places.) x2 > *²<

Answer 1

Answer

We are given the data to calculate the sample variance, s2.

n = 7: 1.2, 3.4, 1.5, 2.1, 3.4, 2.7, 2.7

The sample variance is given by,

$s^2=\frac{1}{n}\sum x_i^2-(\frac{1}{n}\sum x_i)^2$

= 4.11429 (rounded to 5 decimal places)

Next, we have to construct a 95% confidence interval for the population variance $\sigma^2$ .

The (1 - $\alpha$ )% confidence interval for the population variance $\sigma^2$ (when $\mu$ is unknown) is given by,

Σ(r; – 7). Στη – Τ) Xa/25η-1 xi-a/27n-1

Here, $\alpha$ = 0.05, n = 7.

Therefore, $\chi_{0.025;6}^2$ = 14.449 and $\chi_{0.975;6}^2$ = 1.237

And = 7 x 4.11429 = 28.80003

Σ(; – )? = ns?

Therefore, the 95% confidence interval for the population variance $\sigma^2$ is,

(1.99, 23.28) (rounded to 2 decimal places)

Next, we would like to test

$H_0: \sigma^2=0.8$

vs

HQ:02 +0.8

using $\alpha$ = 0.05

The test statistic for $\sigma^2$ (when population mean $\mu$ is unknwon) is,

$\chi^2=\frac{\sum (x_i-\overline{x})^2}{\sigma_0^2}$

Here, $\sum(x_i-\overline{x})^2$ = 28.80003 and $\sigma_0^2$ = 0.8

$\therefore \chi_{obs}^2$ = 28.80003/0.8 = 36 (rounded to 2 decimal places)

Since, here we want to test whether population variance is equal to 0.8 or not, it is a two-sided test.

Therefore, H0 is rejected at 100$\alpha$% significance if

either $\chi_{obs}^2>\chi_{\alpha/2;n-1}^2$

or $\chi_{obs}^2<\chi_{1-\alpha/2;n-1}^2$

and accept otherwise.

Here, $\chi_{0.025;6}^2$ = 14.449 and $\chi_{0.975;6}^2$ = 1.237

Hence, $\chi_{obs}^2>14.45$

or $\chi_{obs}^2<1.24$

Answer: The value of s2 is 4.11429. The 95% confidence interval for the population variance $\sigma^2$ is 1.99 to 23.28. The test statistic $\chi^2$ is 36. The rejection region is $\chi^2$ > 14.45 or $\chi^2$ < 1.24.