Suppose that in a certain region of the country the mean duration of first marriages that end in divorce is 7.8 years, standard deviation 1.2 years. Suppose a sample of 75 divorces were considered **Problem Statement:**Find the probability that in a sample of 75 divorces, the mean duration (age) of the marriages is at most 8 years. That is, if $ X $ represents the age of marriages, find $ P\left( \overline{X} \leq 8 \right) $. (Keep your answer to four decimal places.)---**Explanation:**This problem is asking for the probability that the average duration of marriages in a random sample of 75 divorces is 8 years or less. It involves statistical probability and might require the use of the Central Limit Theorem to solve, which states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large.**Key Terms:**- **Sample Size (n):** 75- **Mean Duration:** The average age of marriages in the sample.- **Probability (P):** The likelihood of the sample mean being less than or equal to 8 years.**Mathematical Representation:**- $ X $: The age of marriages.- $ \overline{X} $: The sample mean.- $ P\left( \overline{X} \leq 8 \right) $: The probability that the sample mean is less than or equal to 8 years.

From the given information, let X be the age of marriages. Number of divorces considered in the…

Answered: Find the probability that in a sample…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that in a certain region of the country the mean duration of first marriages that end in divorce is 7.8 years, standard deviation 1.2 years. Suppose a sample of 75 divorces were considered

**Problem Statement:**

Find the probability that in a sample of 75 divorces, the mean duration (age) of the marriages is at most 8 years. That is, if $ X $ represents the age of marriages, find $ P\left( \overline{X} \leq 8 \right) $. (Keep your answer to four decimal places.)

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**Explanation:**

This problem is asking for the probability that the average duration of marriages in a random sample of 75 divorces is 8 years or less. It involves statistical probability and might require the use of the Central Limit Theorem to solve, which states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large.

**Key Terms:**

- **Sample Size (n):** 75
- **Mean Duration:** The average age of marriages in the sample.
- **Probability (P):** The likelihood of the sample mean being less than or equal to 8 years.

**Mathematical Representation:**

- $ X $: The age of marriages.
- $ \overline{X} $: The sample mean.
- $ P\left( \overline{X} \leq 8 \right) $: The probability that the sample mean is less than or equal to 8 years.

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 + ...

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