41 random samples of monthly electric bill amounts are selected from a normally distributed population. The samples have a mean of $108 and a standard deviation of $5. Construct a 98% confidence interval for the population standard deviation.

A) $1.89 < σ < $2.75
B) $3.14 < σ < $9.02
C) $3.96 < σ < $6.72
D) $4.23 < σ < $6.14

1 answer

To construct a confidence interval for the population standard deviation using the Chi-Square distribution, we can use the formula:

\[ \left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}} \right) \]

Where:

  • \( n \) is the sample size.
  • \( s \) is the sample standard deviation.
  • \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) are the critical values from the Chi-Square distribution for \( n-1 \) degrees of freedom.

In this case:

  • \( n = 41 \)
  • \( s = 5 \)
  • The confidence level is 98%, so \( \alpha = 0.02 \) and \( \alpha/2 = 0.01 \).

The degrees of freedom \( n-1 \) is \( 41 - 1 = 40 \).

Next, we look up the critical values in the Chi-Square distribution table:

  1. For \( \chi^2_{0.01} \) with \( 40 \) degrees of freedom.
  2. For \( \chi^2_{0.99} \) with \( 40 \) degrees of freedom.

Using a Chi-Square table or calculator:

  • \( \chi^2_{0.01}(40) \approx 63.691 \)
  • \( \chi^2_{0.99}(40) \approx 19.493 \)

Now we apply these values to the confidence interval formula:

  1. Calculate the lower limit: \[ \sqrt{\frac{(41-1)(5^2)}{63.691}} = \sqrt{\frac{40 \cdot 25}{63.691}} = \sqrt{\frac{1000}{63.691}} \approx \sqrt{15.69} \approx 3.96 \]

  2. Calculate the upper limit: \[ \sqrt{\frac{(41-1)(5^2)}{19.493}} = \sqrt{\frac{40 \cdot 25}{19.493}} = \sqrt{\frac{1000}{19.493}} \approx \sqrt{51.25} \approx 7.16 \]

Thus, the 98% confidence interval for the population standard deviation is approximately:

\[ 3.96 < \sigma < 7.16 \]

The interval that best fits our results is C) \( 3.96 < \sigma < 6.72 \).