Question

Use the information in Example \(7.8\), but use a sample size of \(55\) to answer the following questions.

a. Find \(P(\bar{x}<7)\).

b. Find \(P(\sum x > 170)\).

c. Find the \(80th\) percentile for the mean of \(55\) scores.

Short answer

Part a. \(P(\bar{x}<7)=0=1\)

Part b. \(P(\sumx > 170)=0.2789\)

Part c. \(80th\) percentile \(=3.13\)

Step-by-step solution

Part a. Step 1. Given information

Lowest stress score \(=1\)

Highest stress score \(=5\)

Sample size \(=55\)

Part a. Step 2. Calculation

\(\mu=\frac{1+5}{2}=3\)

\(\sigma_{x}=\sqrt{\frac{(5-1)^{2}}{12}}=1.15\)

\(\bar{X}=N(3\frac{1.15}{\sqrt{55}}\)

\(P(\bar{x}<7)=normalcdf(1,7,3,\frac{1.15}{\sqrt{55}}\)

\(P(\bar{x}<7)=0=1\)

Part b. Step 1. Calculation

\(\sum X=N((n)(\mu_{x}),(\sqrt{n})(\sigma_{x}))\)

\(P(\sum x>170)=normalcdf(170,E99,(55)(3),(\sqrt{55})(1.15))\)

\(P(\sum x>170)=0.2789\)

Part c. Step 1. Calculation

\(80th\)percentile\(=invNorm(\frac{80}{100},3,\frac{1.15}{\sqrt{55}}\)

\(80th\)percentile\(=3.13\)