An anthropologist wishes to estimate the average height of...

Name: Problem 43, Chapter 7: Mathematical Statistics with Applications
Uploaded: 2021-07-20T18:47:47-08:00
Duration: 2 min 10 s
Channel: Marc Lauzon
Description: An anthropologist wishes to estimate the average height of men for a certain race of people. If the population standard deviation is assumed to be 2.5 inches and if she randomly samples 100 men, find the probability that the difference between the sample mean and the true population mean will not exceed .5 inch.

An anthropologist wishes to estimate the average height of men for a certain race of people. If the population standard deviation is assumed to be 2.5 inches and if she randomly samples 100 men, find the probability that the difference between the sample mean and the true population mean will not exceed .5 inch.

Key Concepts

Central Limit Theorem

This theorem states that regardless of the population's distribution, the distribution of the sample mean approximates a normal distribution when the sample size is sufficiently large. This principle is fundamental when making inferences about population parameters based on sample statistics, as it allows the use of normal probability methods even if the population distribution is not normal.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means from samples of the same size drawn from the population. This concept is key in statistics as it forms the basis for estimating how sample means deviate from the true population mean, aiding in the calculation of probabilities and confidence intervals.

Standard Error

The standard error quantifies the variability of the sample mean, measuring how much the sample mean of a set of observations is expected to differ from the actual population mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it decreases as the sample size increases, leading to more precise estimates.

Standardization (z-Score)

Standardization, often expressed through the z-score, involves converting a raw score into a standard score that reflects its distance from the mean in units of standard deviations. This technique allows statisticians to compare different data sets and calculate probabilities by referencing the standard normal distribution, thereby facilitating the assessment of how unusual or typical a particular sample statistic is under the assumed model.

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