Question

a simple random sample of size n is drawn from a population that is known to be normally distributed. the sample variance, s², is determined to be 12. complete parts (a) through (c).
(a) construct a 90% confidence interval for σ² if the sample size, n, is 20.
the lower bound is 7.56. (round to two decimal places as needed.)
the upper bound is 22.54. (round to two decimal places as needed.)
(b) construct a 90% confidence interval for σ² if the sample size, n, is 30.
the lower bound is 8.18. (round to two decimal places as needed.)
the upper bound is □. (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for confidence interval of variance

The formula for a confidence interval for the population variance \(\sigma^{2}\) is \(\frac{(n - 1)s^{2}}{\chi_{\alpha/2}^{2}}<\sigma^{2}<\frac{(n - 1)s^{2}}{\chi_{1-\alpha/2}^{2}}\)

For a \(90\%\) confidence interval, \(\alpha=1 - 0.90 = 0.10\), and \(\alpha/2=0.05\), \(1-\alpha/2 = 0.95\)

Given \(n = 20\), \(s^{2}=12\)

The degrees of freedom \(df=n - 1=19\)

From the \(\chi^{2}\) - distribution table, \(\chi_{0.05,19}^{2}=30.144\), \(\chi_{0.95,19}^{2}=10.117\)

Step2: Calculate the lower and upper bounds

The lower bound: \(\frac{(n - 1)s^{2}}{\chi_{\alpha/2}^{2}}=\frac{(20 - 1)\times12}{30.144}=\frac{19\times12}{30.144}=\frac{228}{30.144}\approx7.56\)

The upper bound: \(\frac{(n - 1)s^{2}}{\chi_{1-\alpha/2}^{2}}=\frac{(20 - 1)\times12}{10.117}=\frac{19\times12}{10.117}=\frac{228}{10.117}\approx22.54\)

For part (b), \(n = 30\), \(df=n - 1=29\), \(\alpha = 0.10\), \(\alpha/2=0.05\), \(1-\alpha/2=0.95\)

From the \(\chi^{2}\) - distribution table, \(\chi_{0.05,29}^{2}=42.557\), \(\chi_{0.95,29}^{2}=17.708\)

The lower bound: \(\frac{(n - 1)s^{2}}{\chi_{\alpha/2}^{2}}=\frac{(30 - 1)\times12}{42.557}=\frac{29\times12}{42.557}=\frac{348}{42.557}\approx8.18\)

The upper bound: \(\frac{(n - 1)s^{2}}{\chi_{1-\alpha/2}^{2}}=\frac{(30 - 1)\times12}{17.708}=\frac{29\times12}{17.708}=\frac{348}{17.708}\approx19.65\)

Answer:

(a) Lower bound: \(7.56\), Upper bound: \(22.54\)

(b) Lower bound: \(8.18\), Upper bound: \(19.65\)