First, we need to calculate the mean from the data.
[tex]\mu=\frac{\sum ^{}_{}x_i}{n}=\frac{2.5+1.75+2.25+2.67}{4}=\frac{9.17}{4}=2.3[/tex]where n is the number of data points.
Next, we need to calculate the standard deviation of the sample, as follows:
[tex]\begin{gathered} \sigma=\sqrt[]{\frac{\sum^{}_{}(x_i-\mu)^2}{n-1}} \\ \sigma=\sqrt[]{\frac{(2.5-2.3)^2+(1.75-2.3)^2+(2.25-2.3)^2+(2.67-2.3)^2}{4-1}} \\ \sigma=\sqrt[]{\frac{0.04+0.3025+0.0025+0.1369}{3}} \\ \sigma=\sqrt[]{0.1606} \\ \sigma=0.4 \end{gathered}[/tex]a. Using the 68 95 99 rule (see picture above), 68% of neck sizes are in the next range:
μ - σ to μ + σ
2.3 - 0.4 to 2.3 + 0.4
1.9 to 2.7
b. Similarly, 95% of neck sizes are in the next range:
μ - 2σ to μ + 2σ
2.3 - 2*0.4 to 2.3 + 2*0.4
1.5 to 3.1
c. Similarly, 99.7% of neck sizes are in the next range:
μ - 3σ to μ + 3σ
2.3 - 3*0.4 to 2.3 + 3*0.4
1.1 to 3.5
d. The z-score is computed as follows:
[tex]z=\frac{x-\mu}{\sigma}[/tex]where x is the raw score. Substituting with x = 3.3, μ = 2.3, and σ = 0.4, we get:
[tex]z=\frac{3.3-2.3}{0.4}=2.5[/tex]Given that this z-score is greater than 2.32 (the critical value) it is significant at 0.01 level
