7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
4:05 minutesProblem 5.4.29
Textbook Question
Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results.
Dow Jones Industrial Average From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 32 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 700? Assume sigma=1540
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Step 1: Identify the key parameters of the problem. The population mean (μ) is 652, the population standard deviation (σ) is 1540, and the sample size (n) is 32. We are tasked with finding the probability that the sample mean (x̄) is between 400 and 700.
Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is SE = σ / √n. Substitute the given values for σ and n into the formula to compute SE.
Step 3: Convert the sample mean values (400 and 700) into z-scores using the formula z = (x̄ - μ) / SE. Compute the z-scores for both 400 and 700 by substituting the respective values of x̄, μ, and SE.
Step 4: Use the standard normal distribution table (or a statistical software) to find the probabilities corresponding to the z-scores calculated in Step 3. These probabilities represent the cumulative probabilities up to the z-scores.
Step 5: Subtract the smaller cumulative probability (corresponding to the z-score for 400) from the larger cumulative probability (corresponding to the z-score for 700). This difference gives the probability that the sample mean is between 400 and 700.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution
A sampling distribution is the probability distribution of a statistic obtained by selecting random samples from a population. It describes how the sample mean varies from sample to sample. The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution.
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Standard Error
The standard error (SE) measures the dispersion of the sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size (SE = sigma / √n). A smaller standard error indicates that the sample mean is likely to be closer to the population mean, which is crucial for determining probabilities in sampling distributions.
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Z-Score
A Z-score represents the number of standard deviations a data point is from the mean of a distribution. In the context of sampling distributions, it is used to calculate the probability of a sample mean falling within a certain range. The Z-score is calculated using the formula Z = (X - μ) / SE, where X is the sample mean, μ is the population mean, and SE is the standard error.
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