Answered: In what range would you expect to find…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Related questions

Concept explainers

Question

The average number of Hate Crimes in the United States per year (1991 to 2018) is approximately 7900 per year with a standard deviation of approximately 1200 hate crimes per year.

In what range would you expect to find the middle 95% of hate crimes per year?
Between Incorrect and Incorrect.

If you were to draw a sample of 10 years worth of hate crimes, in what range would you expect to find the middle 95% of average number of Hate Crimes per year in this sample?
Between Incorrect a

Expert Solution

Step 1

The random variable X is defined as the number of Hate Crimes in the United States which follows normal distribution with mean 7,900, and standard deviation 1,200.

The range of the middle 95% of Hate Crimes per year in this sample is,

$\begin{array}{rcl} P (x_{1} < X < x_{2}) & = & 0.95 \\ P (\frac{x_{1} - 7, 900}{1, 200} < \frac{X - μ}{σ} < \frac{x_{2} - 7, 900}{1, 200}) & = & 0.95 \\ P (\frac{x_{1} - 7, 900}{1, 200} < z < \frac{x_{2} - 7, 900}{1, 200}) & = & 0.95 \end{array}$

For

$\begin{array}{rcl} (1 - α) & = & 0.95 \\ α & = & 0.05 \\ \frac{α}{2} & = & 0.025 \\ 1 - \frac{α}{2} & = & 0.975 \end{array}$

The probability 0.975 would be converted to z score can be obtained using the excel formula “=NORM.S.INV(0.975)”. The z score value is 1.96.

The probability 0.025 would be converted to z score can be obtained using the excel formula “=NORM.S.INV(0.025)”. The z score value is -1.96.

The required value is,

$\begin{array}{rcl} P (\frac{x_{1} - 7, 900}{1, 200} < z < \frac{x_{2} - 7, 900}{1, 200}) & = & 0.95 \\ P (\frac{x_{1} - 7, 900}{1, 200} < z < \frac{x_{2} - 7, 900}{1, 200}) & = & P (- 1.96 < z < 1.96) \end{array}$

Comparing on both sides.

$\begin{array}{rcl} \frac{x_{1} - 7, 900}{1, 200} & = & - 1.96 a n d \frac{x_{2} - 7, 900}{1, 200} = 1.96 \\ x_{1} - 7, 900 & = & - 2, 352 a n d x_{2} - 7, 900 = 2, 352 \\ x_{1} & = & 7, 900 - 2, 352 a n d x_{2} = 7, 900 + 2, 352 \\ x_{1} & = & 5, 548 a n d x_{2} = 10, 252 \end{array}$

Thus, the range of the middle 95% of hate crimes per year is between 5,548 and 10,252.

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Continuous Probability Distribution$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.