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Sampling Distribution. Sampling Distribution of t he Sampling Mean. In inferential statistics, we want to use characteristics of the sample to estimate the characteristics of the population. Example….
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Sampling Distribution of the Sampling Mean In inferential statistics, we want to use characteristics of the sample to estimate the characteristics of the population. Example… A large tank of fish from a hatchery is being delivered to the lake. We want to know the average length of the fish in the tank. Instead of measuring all the fish, we randomly sample some of them and use the sample mean to estimate the population mean.
Note… The sample mean denoted x̅ is random since its value depends on the sample chosen. It is called a statistic. The population mean is fixed, usually denoted as µ.
Example…. The population is the weight of six pumpkins (in pounds) displayed in a carnival "guess the weight" game booth. You are asked to guess the average weight of the six pumpkins by taking a random sample without replacement from the population. First: Lets make a table
Now lets Calculate our Population Mean µ µ = (19 + 14 + 15 + 9 + 16 + 17) / 6 = 15 pounds Next let’s obtain the sampling distribution of the sample mean for a sample size of 2 when one samples without replacement
We can use a table.. Find the mean of the sample mean when the sample size is 2: Mean of sample mean = (16.5 + 17.0 + 14.0 + 17.5 + 18.0 + 14.5 + 11.5 + 15.0 + 15.5 + 12.0 + 15.5 + 16.0 + 12.5 + 13.0 + 16.5) / 15 = 15 pounds
Let’s obtain the sampling distribution for the sample mean when the sample size is 5. Mean of sample mean = (14.6 + 14.8 + 16.2 + 15.0 + 15.2 + 14.2) / 6 = 15 pounds
The following dot plots show the distribution of the sample means corresponding to sample sizes of 2 and of 5. • Sampling error: is the error resulting from using a sample characteristic to estimate a population characteristic. • Sample size and sampling error: As the dot plot above shows, the possible sample means cluster more closely around the population mean as the sample size increases. Thus, possible sampling error decreases as sample size increases.
The mean of sample mean is the population mean. That is: When sampling with replacement, the standard deviation of the sample mean equals the population standard deviation divided by the square root of the sample size .
Central Limit Theorem For a large sample size (rule of thumb: n ≥ 30), is approximately normally distributed, regardless of the distribution of the population one samples from. If the population has mean µ and standard deviation σ then has mean µ and standard deviation .
Example… The engines made by Ford for speedboats had an average power of 220 horsepower (HP) and standard deviation of 15 HP. A potential buyer intends to take a sample of four engines and will not place an order if the sample mean is less than 215 HP. What is the probability that the buyer will not place an order? Want to find P( < 215) = ?
Answer: We need to know whether the distribution of the population is normal since the sample size is too small: n = 4 (less than 30 which is required in the central limit theorem). If someone confirms that the population normal, then we can proceed since the sampling distribution of the mean of a normal distribution is also normal for all sample sizes. If the population follows a normal distribution, we can conclude that has a normal distribution with mean 220 HP and a standard deviation of .
If the customer just samples four engines, the probability that the customer will not place an order is 25.14%.
Math 4, Unit 1 learning task How long are the words in the Gettysburg Address?
GPS Math IV Standards MM4D1. Using simulation, students will develop the idea of the central limit theorem. MM4D2. Using student-generated data from random samples of at least 30 members, students will determine the margin of error and confidence interval for a specified level of confidence. MM4D3. Students will use confidence intervals and margin of error to make inferences from data about a population. Technology is used to evaluate confidence intervals, but students will be aware of the ideas involved.
Observations What do you notice about the shape of the distributions?
Summary How do the means and standard deviations of the samples compare to the overall mean (μ) and standard deviation (σ) ?
Central Limit Theorem When the sample size (n) is large and taken as independent random variables, the sampling distribution will be approximately normally distributed.
