6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
5:35 minutesProblem 5.4.40
Textbook Question
In Exercises 39 and 40, determine whether the finite correction factor should be used. If so, use it in your calculations when you find the probability.
Old Faithful In a sample of 100 eruptions of the Old Faithful geyser at Yellowstone National Park, the mean interval between eruptions was 129.58 minutes and the standard deviation was 108.54 minutes. A random sample of size 30 is selected from this population. What is the probability that the mean interval between eruptions is between 120 minutes and 140 minutes?
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Determine whether the finite population correction factor (FPC) should be used. The FPC is applied when the sample size (n) is greater than 5% of the population size (N). If the population size is not provided, assume it is large enough to ignore the FPC unless otherwise stated.
Identify the parameters of the problem: the population mean (μ = 129.58 minutes), the population standard deviation (σ = 108.54 minutes), the sample size (n = 30), and the range of interest for the sample mean (120 minutes to 140 minutes).
Calculate the standard error of the mean (SE). The formula for SE is: . Substitute the values of σ and n into the formula.
Standardize the range of interest (120 to 140 minutes) into z-scores using the formula: , where X is the value of interest, μ is the population mean, and SE is the standard error calculated in the previous step.
Use the z-scores to find the cumulative probabilities from the standard normal distribution table. Subtract the cumulative probability corresponding to the lower z-score (120 minutes) from the cumulative probability corresponding to the upper z-score (140 minutes) to find the probability that the sample mean falls within the given range.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Finite Correction Factor
The Finite Correction Factor (FCF) is used in statistics when sampling without replacement from a finite population. It adjusts the standard error of the sample mean to account for the fact that the sample size is a significant fraction of the total population. This is important when the sample size is more than 5% of the population, as it helps to provide a more accurate estimate of variability.
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Central Limit Theorem
The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n ≥ 30). This theorem is crucial for calculating probabilities related to sample means, as it allows us to use normal distribution properties even when the original data is not normally distributed.
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Standard Error of the Mean
The Standard Error of the Mean (SEM) quantifies the amount of variability in the sample mean estimates from the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. A smaller SEM indicates that the sample mean is a more precise estimate of the population mean, which is essential for determining probabilities in hypothesis testing and confidence intervals.
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