Question

1. (25 Points) The following are neck-length measurements taken of Zoo Giraffes around the United States (in meters): Xi = 2.50 X = 175 X3 = 2.25 X = 2.67 a. Assuming this data is (roughly) uniform in distribution, between which range of neck sizes would you estimate 68% of zoo giraffes in the U.S. to fall? b. Between which range of neck sizes would you estimate 95% of zoo giraffes in the U.S. to fall? 1.9 and 2.68 Between which range of neck sizes would you estimate 99.7% of zoo giraffes in the U.S: to fall? d. Calculate the Z-score for a zoo giraffe with a neck length of 3.30 meters and determine if it is significant. C.

Answer 1

First, we need to calculate the mean from the data.

`\mu=\frac{\sum ^{}_{}x_i}{n}=(2.5+1.75+2.25+2.67)/(4)=(9.17)/(4)=2.3`

where n is the number of data points.

Next, we need to calculate the standard deviation of the sample, as follows:

`\begin{gathered} \sigma=\sqrt[]{\frac{\sum^{}_{}(x_i-\mu)^2}{n-1}} \\ \sigma=\sqrt[]{((2.5-2.3)^2+(1.75-2.3)^2+(2.25-2.3)^2+(2.67-2.3)^2)/(4-1)} \\ \sigma=\sqrt[]{(0.04+0.3025+0.0025+0.1369)/(3)} \\ \sigma=\sqrt[]{0.1606} \\ \sigma=0.4 \end{gathered}`

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:

`z=(x-\mu)/(\sigma)`

where x is the raw score. Substituting with x = 3.3, μ = 2.3, and σ = 0.4, we get:

`z=(3.3-2.3)/(0.4)=2.5`

Given that this z-score is greater than 2.32 (the critical value) it is significant at 0.01 level