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Statistics - Sampling

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    Study Guide Statistics – Sampling 1. Quiz: Sampling Distributions 1. Question The best description of the sampling distribution of a sample statistic is: Answer Choices • the distribution of the values of the statistic for all individuals in the sample. • the distribution of the values of the statistic for some samples, with the same size, selected from the population. • the distribution of the values of the statistic for all possible samples, with the same size, selected from the population. Correct Answer the distribution of the values of the statistic for all possible samples, with the same size, selected from the population. Why This Is Correct A sampling distribution is formed by taking all possible random samples of the same size from a population and calculating the statistic (like the mean) for each sample. This shows how the statistic behaves across repeated sampling. 2. Question Sampling variability means that: Answer Choices • the value of the sample statistic will vary for the samples. • the sample statistic will vary from the population parameter. • the value of the sample statistic will vary for different sample sizes.

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    Study Guide Correct Answer the value of the sample statistic will vary for the samples. Why This Is Correct Even when all samples come from the same population and have the same size , the statistic (like a mean or proportion) will not be exactly the same each time. This natural change from sample to sample is called sampling variability . 3. Question Which is not true for the mean of the sampling distribution? Answer Choices • It is the mean of the statistic for all of the samples in the distribution. • It is the same as the population parameter. • It depends on the sample size. Correct Answer It depends on the sample size. Why This Is Correct The mean of the sampling distribution of the sample mean is: equal to the population mean ( μ) This value does not change when the sample size changes. What does change with sample size is the spread (standard deviation) of the sampling distribution. 2 . Random and Systematic Error (Bias) When we use a sample to estimate something about a whole population , our answer may not be perfect.

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    Study Guide This happens because two main types of error can occur: 1. Random error (sampling variability) 2. Systematic error (bias) Let’s understand both in a simple way. 1) Random Error (Sampling Variability) Random error happens naturally because samples can differ from each other. Even if we take samples correctly, the results may change a little each time. What it looks like: • Some sample results will be below the true population value. • Some sample results will be above the true population value. • This happens just because of chance in sampling. Example idea (from the chapter): In the previous section, ten sample means were taken. They were not all the same as the true population mean : • Some sample means were smaller • Some were larger Even the average of those ten sample means was a little lower than the true population mean. But if we took ten new samples again , the average of those new sample means might be higher than the true population mean. Key point: Random error causes values to scatter around the true value , but not always equally each time. 2) Systematic Error (Bias) Systematic error (also called bias) happens when the method of sampling is unfair. Instead of being “sometimes high and sometimes low,” it pushes results in one direction repeatedly .

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    Study Guide What bias means: • The sample results consistently overestimate the true value, or • The sample results consistently underestimate the true value Example of bias: Suppose your list of magazine subscribers comes from a database of air travelers . That would create a problem because: • People with lower income are less likely to travel by air • Many lower - income subscribers might not even be included in the list • So they won’t have a chance to be selected in the sample As a result, the sample would likely give an income average that is too high compared to the true average income of all subscribers . Key point: Bias shifts results away from the true value in a consistent direction . Understanding Figure 1 (Dot Plots) Figure 1 shows how random error and bias affect sample results. Each dot plot shows where the sample results fall compared to the true population value . Important observation: • The two dot plots on the right show systematic error (bias) Their dots are not centered near the true value. • The two dot plots on the left are centered close to the true value These do not show bias. Simple rule from the figure: • Center close to true value = low bias • Center far from true value = high bias • Dots spread out a lot = high variability • Dots tightly grouped = low variability

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    Study Guide Figure 1.Random (sampling) error and systematic error (bias) distort the estimation of population parameters from sample statistics. 3. Central Limit Theorem When we take samples from a population, we often study the sampling distribution of the sample mean (written as ( x̄ ). This helps us understand how sample means behave when we take many samples of the same size. 3.1 What if the population is normal? If the population of all magazine subscribers is normally distributed , then the sampling distribution of the sample means will also be normal . This makes working with probabilities easier, because we can use the normal curve right away. 3.2 What if the population is NOT normal? This is where the Central Limit Theorem (CLT) becomes very important.

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    Study Guide The Central Limit Theorem says: Even if the population distribution is strongly non - normal , the sampling distribution of the mean will become approximately normal as long as the sample size is large enough . A common rule is: If ( n > 30 ), the sampling distribution of the mean is approximately normal. Because of this, we can still use normal probability methods (like Z - scores and the normal curve) for large samples , even if the original population is not normal. 3.3 Mean of the Sampling Distribution The CLT also tells us something powerful about the average of all sample means . The mean of the sampling distribution of sample means is: This means: • The average of all sample means equals the population mean. • So, the sample mean is an unbiased estimator of the population mean. 3.4 Standard Deviation of the Sampling Distribution (Standard Error) The standard deviation of the sampling distribution of sample means is: This value is also called the: Standard Error of the Mean It tells us how much the sample mean ( \ bar{x} ) typically varies from sample to sample.

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    Study Guide Important idea: • Bigger samples → smaller standard error • That means sample means become more consistent when the sample size increases. Why does the sample size matter? As the sample size ( n ) increases: • The sample mean becomes less variable • The sampling distribution becomes narrower • We get more reliable averages In simple words: The mean of many observations is less variable than the mean of only a few observations. Standard Error (A Quick Note) Every statistic has its own standard error , which measures the statistic’s random variability . For the sample mean: • The standard error tells us how “spread out” the sample means are. Example 1 Problem If the population mean number of fish caught per trip is: • Population mean: ( μ = 3.2 ) • Population standard deviation: ( σ = 1.8 ) Find the mean and standard deviation of the sampling distribution for samples of size: • ( n = 40 )

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    Study Guide Step 1: Find the mean of the sampling distribution So the mean of the sampling distribution is: Step 2: Find the standard deviation (standard error) So the standard deviation of the sampling distribution is: ( σ x̄ = 0.285 ) Final Answer for Example 1 • Mean of the sampling distribution: ( μ x̄ = 3.2 ) • Standard deviation (standard error): ( σ x̄ = 0.285) 4. Quiz: Central Limit Theorem 1. Question The Central Limit Theorem says that the sampling distribution of the sample mean is approximately normal if:

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    Study Guide Answer Choices • all possible samples are selected. • the sample size is large. • the standard error of the sampling distribution is small. Correct Answer the sample size is large. Why This Is Correct The Central Limit Theorem (CLT) states that when the sample size is large , the sampling distribution of the sample mean becomes approximately normal , even if the original population is not normal. 2. Question The Central Limit Theorem says that the mean of the sampling distribution of the sample means is: Answer Choices • equal to the population mean divided by the square root of the sample size. • close to the population mean if the sample size is large. • exactly equal to the population mean. Correct Answer exactly equal to the population mean. Why This Is Correct The mean of the sampling distribution of ( \ bar{x}) is: μ ₓ̄ = μ So the average of all sample means is exactly the population mean .

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    Study Guide 3. Question The Central Limit Theorem says that the standard deviation of the sampling distribution of the sample means is: Answer Choices • equal to the population standard deviation divided by the square root of the sample size. • close to the population standard deviation if the sample size is large. • exactly equal to the standard deviation. Correct Answer equal to the population standard deviation divided by the square root of the sample size. Why This Is Correct The standard deviation of the sampling distribution of ( \ bar{x}) is called the standard error : σ x̄ = σ / √ n It becomes smaller as sample size increases . 4. Question Samples of size 25 are selected from a population with mean 40 and standard deviation 7.5 . The mean of the sampling distribution of sample means is: Answer Choices • 7.5 • 8 • 40 Correct Answer 40

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