Question

the mayor of a small town wants to estimate the average property value for the houses built in the last year. he randomly selects 15 houses and pays an appraiser to determine the value of each house. the mean value of these houses is $183,100 with a standard deviation of $29,200. calculate a 90% confidence interval for the mean value of new houses in this town. assume the conditions are met. (round each bound to the nearest whole number.)

Explanation:

Step1: Identify critical values

Sample size $n=15$, so degrees of freedom $df = n-1 = 14$. For 90% confidence, the critical t-value $t_{\alpha/2,14} = 1.761$ (from t-table).

Step2: Compute standard error

Standard error $SE = \frac{s_x}{\sqrt{n}} = \frac{29200}{\sqrt{15}}$
$SE = \frac{29200}{3.87298} \approx 7539.38$

Step3: Calculate margin of error

Margin of error $E = t_{\alpha/2} \times SE$
$E = 1.761 \times 7539.38 \approx 13276.85$

Step4: Find confidence interval bounds

Lower bound: $\bar{x} - E = 183100 - 13276.85 = 169823.15$
Upper bound: $\bar{x} + E = 183100 + 13276.85 = 196376.85$

Step5: Round to whole numbers

Lower bound $\approx 169823$, Upper bound $\approx 196377$

Answer:

169823 to 196377