Answered: CNNBC recently reported that the mean…

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...

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Question

CNNBC recently reported that the mean annual cost of auto insurance is 972 dollars. Assume the standard deviation is 102 dollars. You will use a simple random sample of 58 auto insurance policies.

Find the probability that a single randomly selected policy has a mean value between 977.4 and 997.4 dollars.
P(977.4 < X < 997.4) =

Find the probability that a random sample of size n=58 has a mean value between 977.4 and 997.4 dollars.
P(977.4 < ¯x < 997.4) =

Enter your answers as numbers accurate to 4 decimal places.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

$Let us denote : X = the mean annual cost of auto insurance X follows a normal distribution with mean = 972 and standard deviation = 102 n = sample size = 58 \bar{X} follows a normal distribution with mean = 972 and standard deviation = \frac{102}{\sqrt{58}} = 13.3932561517$

Step 2

$A) The probability that a single randomly selected policy has a mean value between 977.4 and 997.4 dollars is : P (977.4 < X < 997.4) = P (\frac{977.4 - 972}{102} < Z < \frac{997.4 - 972}{102}) = P (0.0529411764706 < Z < 0.249019607843) = 0.0772$

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