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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 6 - Normal Probability DistributionsProblema 20b
Capitolo 6, Problema 20b

Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with μ = 95.5 and σ = 16.0 (based on data from “Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm,” by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected without replacement for a follow-up study.


b. Find the probability that the mean IQ score of the follow-up sample is between 95 and 105.

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Step 1: Identify the key parameters of the problem. The population mean (μ) is 95.5, the population standard deviation (σ) is 16.0, the sample size (n) is 50, and the population size (N) is 275. The goal is to find the probability that the sample mean IQ score is between 95 and 105.
Step 2: Adjust the standard error of the mean to account for the finite population correction factor. The formula for the corrected standard error is: σx̄ = σ1n × N - nN - 1, where N is the population size, n is the sample size, and σ is the population standard deviation.
Step 3: Calculate the z-scores for the sample mean values of 95 and 105. The z-score formula is: z = x̄ - μσx̄, where x̄ is the sample mean, μ is the population mean, and σx̄ is the corrected standard error of the mean.
Step 4: Use the z-scores to find the cumulative probabilities corresponding to the sample mean values of 95 and 105. This can be done using a standard normal distribution table or statistical software. The cumulative probability for a z-score represents the area under the standard normal curve to the left of that z-score.
Step 5: Subtract the cumulative probability for the lower z-score (corresponding to 95) from the cumulative probability for the higher z-score (corresponding to 105). This difference gives the probability that the sample mean IQ score is between 95 and 105.

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Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this context, the IQ scores of the children are said to approximate a normal distribution, which allows for the application of statistical methods to calculate probabilities related to the mean and standard deviation.
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Finding Standard Normal Probabilities using z-Table

Sampling Distribution of the Mean

The sampling distribution of the mean refers to the distribution of sample means obtained from all possible samples of a specific size from a population. When sampling without replacement, the mean of the sample will tend to be normally distributed around the population mean, especially as the sample size increases, which is crucial for calculating probabilities regarding the sample mean.
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Sampling Distribution of Sample Proportion

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the samples are independent. This theorem is essential for determining the probability that the mean IQ score of the follow-up sample falls within a specified range, as it justifies the use of normal distribution properties for the sample mean.
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Calculating the Mean
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Sampling Distribution of the Sample Mean


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