Question

unhealthy days in cities the number of unhealthy days based on the aqi (air quality index) for a random sample of metropolitan areas is shown. round the sample statistics and final answers to at least one decimal place. 23 10 32 24 0 5 27 5 50 6 send data to excel construct a 95% confidence interval based on the data. assume the variable is normally distributed. \\(\square < \mu < \square\\)

Explanation:

Step1: Calculate sample mean

Data: 0,5,5,6,10,23,24,27,32,50. Sum = 0+5+5+6+10+23+24+27+32+50 = 182. Mean $\bar{x} = \frac{182}{10} = 18.2$.

Step2: Calculate sample standard deviation

Deviations squared: $(0-18.2)^2=331.24$, $(5-18.2)^2=174.24$, $(5-18.2)^2=174.24$, $(6-18.2)^2=148.84$, $(10-18.2)^2=67.24$, $(23-18.2)^2=23.04$, $(24-18.2)^2=33.64$, $(27-18.2)^2=77.44$, $(32-18.2)^2=190.44$, $(50-18.2)^2=1011.24$. Sum of squares = 331.24+174.24+174.24+148.84+67.24+23.04+33.64+77.44+190.44+1011.24=2231.6. Sample variance $s^2 = \frac{2231.6}{10-1} \approx 247.96$. Sample standard deviation $s \approx \sqrt{247.96} \approx 15.75$.

Step3: Find t-critical value

df = 10-1=9, 95% confidence. t-value $t_{0.025,9}=2.262$.

Step4: Compute margin of error

$E = t \times \frac{s}{\sqrt{n}} = 2.262 \times \frac{15.75}{\sqrt{10}} \approx 2.262 \times 4.98 \approx 11.27$.

Step5: Construct confidence interval

Lower: $18.2 - 11.27 = 6.93$. Upper: $18.2 + 11.27 = 29.47$. Rounded to 1 decimal: 6.9 and 29.5.

Answer:

6.9 < μ < 29.5