Statistics Chapter 7

Questions and Answers

What is sampling error?

Sampling Error is the error resulting from using a sample to estimate a population characteristic.

Define the sampling distribution of the sample mean.

The sampling distribution of the sample mean is the distribution of all possible sample means of a given sample size.

Compare the sampling error from small samples with the sampling error of large samples.

The sampling error of large samples tends to be less than the sampling error for small samples.

The mean of all possible sample means always equals what?

The population mean. Signup and view all the answers

What happens to the mean of the sampling distribution of the sample mean as sample size gets bigger?

The mean of the sampling distribution of the sample mean approaches the population mean. Signup and view all the answers

What happens to the standard deviation of the sampling distribution of the sample mean as sample size gets bigger?

The standard deviation approaches 0. Signup and view all the answers

Give the equation for the standard deviation of the sample mean.

sigma_xbar = sigma / √n. Signup and view all the answers

If a variable x is normally distributed with mean μ and standard deviation σ, what can we say about the variable x̄ for a sample size n?

The variable x̄ is also normally distributed and has mean μ and standard deviation σ/√n. Signup and view all the answers

What does the Central Limit Theorem (CLT) state?

For a relatively large sample size, the variable x̄ is approximately normally distributed, regardless of the distribution of the variable. Signup and view all the answers

What can we say about the distribution of x̄?

If x is normally distributed, so is x̄, regardless of sample size. If the sample size is large, x̄ is approximately normally distributed, regardless of the distribution of x. Signup and view all the answers

How do you find the probability that a variable will be within a specified range if the variable is normally distributed?

Find Z-scores and the area under the curve between the endpoints of the range. Signup and view all the answers

How do you find the percent of data points that lie within a specified range if the variable is normally distributed?

Find Z-scores and the area under the curve between the endpoints of the range. Signup and view all the answers

If a problem tells you that the error can be within 5, what do you do?

If the error can be within five, then the 'acceptable range' is Mean - 5 to Mean + 5. Signup and view all the answers

Study Notes

Sampling Error

Sampling Error refers to discrepancies that arise when a sample is used to estimate characteristics of a larger population.

Sampling Distribution of the Sample Mean

Defined as the distribution of all conceivable sample means for a specific sample size.

Sampling Error Comparison

Larger samples exhibit reduced sampling error compared to smaller samples.

Population Mean and Sample Means

The mean of all possible sample means consistently equals the population mean.

Standard Deviation of Sample Means

The standard deviation of the sample mean is influenced by sample size, approaching 0 as sample size increases.

Equation for Standard Deviation of Sample Mean

σ̄ = σ / √n
- σ̄ represents the standard deviation of the sample mean.
- σ denotes the population standard deviation.
- n signifies the sample size.

Normally Distributed Variables

If variable x is normally distributed with mean μ and standard deviation σ, then the sample mean x̄ is also normally distributed with:
- Mean μ
- Standard deviation σ/√n

Central Limit Theorem (CLT)

As sample size becomes sufficiently large, the variable x̄ tends toward a normal distribution, regardless of the original variable's distribution. Accuracy improves with larger sample sizes.

Distribution of x̄

If x is normally distributed, x̄ is likewise normally distributed, independent of sample size.
For larger sample sizes, x̄ approximates a normal distribution regardless of x's distribution.

Finding Probabilities in Normal Distribution

To determine the probability of a variable falling within a certain range when normally distributed, calculate Z-scores and measure the area under the curve between endpoints.

Percent of Data Points Within a Range

Similar to probability calculation, find Z-scores to measure the area under the curve for the specified range limits.

Acceptable Error Range

An error margin of within 5 sets an acceptable range from Mean - 5 to Mean + 5. Z-scores may be necessary for these endpoints, along with the area calculation between them.

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Description

Explore the key concepts of Chapter 7 focusing on the Sampling Distribution of the Sample Mean. This quiz includes important definitions and comparisons necessary for mastering the topic. Test your understanding of sampling error and its implications in statistical analysis.