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Magazine Chapter 7: Problem 4
Suppose that samples of size \(n=25\) are selected at random from a normal population with mean 100 and standard deviation \(10 .\) What is the probability that the sample mean falls in the interval from \(\mu_{\bar{X}}-1.8 \sigma_{\bar{X}}\) to \(\mu_{\bar{X}}+1.0 \sigma_{\bar{X}}\) ?
Short Answer
Expert verified
The probability is approximately 0.8054.
Step by step solution
01
Understand the Problem
We need to calculate the probability that the sample mean falls within a specific range around the population mean. The population has a mean of 100 and a standard deviation of 10. The sample size is 25.
02
Identify the Mean of the Sampling Distribution
The mean of the sampling distribution, which is the expected value of the sample mean, is equal to the population mean, \( \mu_{\bar{X}} = \mu = 100 \).
03
Calculate the Standard Deviation of the Sampling Distribution
The standard deviation of the sampling distribution (also known as the standard error) is given by \( \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{25}} = 2 \).
04
Define the Interval
The interval is given by \( \mu_{\bar{X}} - 1.8\sigma_{\bar{X}} \) to \( \mu_{\bar{X}} + 1.0\sigma_{\bar{X}} \). Substituting the values, we have \( 100 - 1.8 \times 2 = 96.4 \) and \( 100 + 1.0 \times 2 = 102 \). Therefore, the interval is [96.4, 102].
05
Use the Standard Normal Distribution
Since the sample mean is normally distributed, we convert the interval into z-scores. This can be done using the formula \( z = \frac{X - \mu_{\bar{X}}}{\sigma_{\bar{X}}} \).
06
Calculate Z-Scores for the Interval
Calculate the z-score for the lower limit: \( z_{1} = \frac{96.4 - 100}{2} = -1.8 \). Calculate the z-score for the upper limit: \( z_{2} = \frac{102 - 100}{2} = 1.0 \).
07
Find the Probability Using Z-Scores
Using a standard normal distribution table or calculator, find the probability: \( P(Z < 1.0) - P(Z < -1.8) \). This gives \( P(Z < 1.0) \approx 0.8413 \) and \( P(Z < -1.8) \approx 0.0359 \). The probability that the sample mean falls within the interval is \( 0.8413 - 0.0359 = 0.8054 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sampling Distribution
When we talk about sampling distribution, we're focusing on how the mean of samples (of a certain size) behaves, rather than individual observations. Imagine taking several samples from a population and calculating the mean for each. The distribution of these means is what we call the sampling distribution.
- The mean of the sampling distribution (\(\mu_{\bar{X}}\)) is equivalent to the mean of the overall population (\(\mu\)).
- Even if the population is not normally distributed, the sampling distribution tends to be normal, especially when the sample size is large, thanks to the Central Limit Theorem.
Normal Distribution
The normal distribution is a fundamental concept in probability theory, characterized by its bell-shaped curve. When data points cluster around a mean value symmetrically, they often follow a normal distribution.
- It is defined by two parameters: the mean (location of the center of the graph) and the standard deviation (how spread out the data is).
- A critical property is that approximately 68% of data lies within one standard deviation of the mean, 95% within two, and 99% within three.
- In our exercise, both the population and the sample mean's distribution are normal, which allows us to use z-scores to find probabilities.
Standard Error
The standard error is essentially the standard deviation of the sampling distribution of a statistic. It provides an estimate of how far the sample mean is likely to be from the true population mean.
- Calculated as the population standard deviation \(\sigma\) divided by the square root of the sample size \(n\): \(\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}\)
- In smaller samples, the standard error will be larger, indicating more variability in the sample mean.
- In our context, it's used to translate a sample standard deviation to that of a whole population, based on a known or assumed population variance.
Z-Score
The z-score is a way to describe a statistic's position relative to the mean of the distribution. It shows how many standard deviations an element lies from the mean.
- Computed by \(z = \frac{X - \mu_{\bar{X}}}{\sigma_{\bar{X}}}\)
- Z-scores help to standardize different datasets, allowing comparisons even if they have different means or standard deviations.
- In probability, using z-scores aids in determining the probability that a data point falls within a particular range (here, within the interval for our sample mean).
