STAT 170 Ch 9 guided notes

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STAT 170 Ch 9 - Inferring Population Means Chapter 9 Topics • Apply the Central Limit Theorem for sample means • Make inferences about population means using confidence intervals and hypothesis tests Section 9.1 Sample Means of Random Samples • Discuss the Accuracy and Precision of the Sample Mean as an Estimate of the Population Mean • Find the Standard Error of the Sampling Distribution of the Sample Mean Three Characteristics of the Sample Mean Notation Review: μ = population mean σ = population standard deviation s = sample standard deviation x = sample mean Accuracy and Precision of the Sample Mean The ______________________ of an estimator is measured by the bias , and the ______________________ is measured by the standard error . The sample mean is unbiased when estimating the population mean – on average, the sample mean is the same as the population mean. The precision of the sample mean depends on the variability in the population. The more observations we collect, the more precise the sample mean becomes. Sampling Distribution of the Sample Mean The sampling distribution can be thought of as the distribution of _________________________________________ that would result from drawing repeated samples of a certain size from the population. Reminder: The standard deviation of the sampling distribution is called the standard error . Example: The response time for the population of all emergency calls to a fire department between 2013 and 2016. It is right-skewed with a mean response time of 6.3 minutes and a standard deviation of 2.8 minutes . Take a simulation that involves a random sample of 50 observations from the data set and calculate the average response time. Repeat this simulation many times: 1. What is the typical value of the sample mean? 2. If unbiased, how far is it from 6.3 minutes?
Effect of Sample Size on the Sampling Distribution As the sample size _________________ the standard error ___________________ but the mean remains the _____. Accuracy and Bias of the Sample Mean For all populations, the sample mean, if based on a _______________ sample, is _______________ when estimating the population mean. The standard error of the sample mean is As the sample size increases, the sample mean becomes more ______________. Example : Playlists A student has a large digital music library. The mean length of the songs is 258 seconds with a standard deviation of 87 seconds. The student creates a playlist that consists of 30 randomly selected songs. a. Is the mean value of 258 seconds a parameter or a statistic? Explain. b. What should the student expect the average song length to be for his playlist? c. What is the standard error for the mean song length of 30 randomly selected songs? In-Class Engagement 1. A study of all the students at a small college showed a mean age of 20.7 and a standard deviation of 2.5 years. a. Are these number statistics or parameters? Explain. b. Label both numbers with their appropriate symbol. 2. Drivers in Alaska drive fewer miles yearly than motorists in any other state. The annual number of miles driven per licensed driver in Alaska is 9134 miles. Assume the SD is 3200 miles. A random sample of 100 licensed drivers in Alaska was selected and the mean number of miles driven yearly for the sample was calculated. a. What value would we expect for the sample mean? b. What is the standard error for the sample? What does this mean? Your Turn 1. A survey of 100 random full-time students at a large university showed the mean number of semester units that students were enrolled in was 15.2 with a standard deviation of 1.5 units. a. Label the numbers with their appropriate symbol.
2. The average shower in the US lasts 8.2 minutes. Assume the standard deviation of 2 minutes. a. Do you expect the shape of the distribution of shower lengths to be normal, right-skewed, or left-skewed? Explain. b. Suppose we survey a random sample of 100 people to find the length of their last shower. We calculate the mean length from the sample and record the value. We repeat this survey 500 times. What will be the shape of the distribution of these sample means? c. Refer to part b. What will be the mean and the standard deviation of the distribution of these sample means? Section 9.2 Central Limit Theorem for Sample Means • Apply the Central Limit Theorem for Sample Means to Calculate Probabilities • Describe the Features of the t -distribution Central Limit Theorem for Sample Means If certain conditions are met, the Central Limit Theorem assures us that the distribution of sample means follows an approximately ___________________ distribution no matter what the shape of the population distribution. *Note: If the population is Normally distributed, then the sampling distribution is normally distributed, regardless of the sample size. Conditions to Check When determining whether you can apply the Central Limit Theorem to analyze data, consider these three conditions: 1. _______________ Sample and _____________________________ . Each observation is collected randomly from the population and observations are independent of each other. 2. ____________ sample . Either the population distribution is Normal or the sample size is large. (usually 25 is large enough). 3. _________ population . If the sample is collected without replacement then the population must be at least 10 times larger than the sample size. If the three conditions are met, then the sampling distribution of _________________________________________ is approximately Visualizing Distributions of the Sample Mean The figure to the right shows the distribution of in-state tuition and fees for all two-year colleges in the United States for the 2012-2013 academic year. Note that it is skewed and multimodal.
These figures show the distribution of the sample mean for samples of size 30 and size 90 drawn from the skewed multimodal distribution on the previous page. Note the shape of each is approximately _____________________ and the standard error ___________________ as the ______________________________ increases. Example: Weights of 10-Year-Old Boys According to data from the National Health and Nutrition Exam Survey, the mean weight of 10-year-old boys is 88.3 pounds with a standard deviation of 2.06 pounds. Assume the distribution of weights is Normal. a. Suppose we take a random sample of 30 boys from this population. Can we find the approximate probability that the average weight of this sample will be above 89 pounds? If so, find it. If not, explain why. b. Suppose we take a random sample of 10 boys from this population. Can we find the approximate probability that the average weight of this sample will be above 89 pounds? If so, find it. If not, explain why not. Example: Home Prices Home prices in a certain community have a distribution that is skewed right. The mean of the home prices is $498,000 with a standard deviation of $25,200. a. Suppose we take a random sample of 30 homes in this community. What is the probability that the mean of this sample is between $500,000 and $510,000? b. Suppose we take a random sample of 10 homes in this community. Can we find the approximate probability that the mean of the sample is more than $510,000? If so, find it. If not, explain why not.
Identify the Distribution There are three distributions below. One of these distributions is a population. The other two distributions are approximate sample distributions of the sample means randomly sampled from the population, one of sample size 10 and the other of sample size 25. Match the graph with the correct distribution. Your Turn 1. Some sources report that the weights of full-term newborn babies have a mean of 7 pounds and a standard deviation of 0.6 pounds and are normally distributed. a. What is the probability that one newborn baby will have a weight within 0.6 pound of the mean? b. Explain why the sampling distribution of sample means with sample size 4 is normally distributed. c. What is the probability the average of four babies’ weights will be within 0.6 pound of the mean? d. Explain the difference between (a) and (c). 2. Some sources report that the weights of full-term newborn babies have a mean of 7 pounds and a standard deviation of 0.6 pounds and are normally distributed. In the given outputs, the shaded areas (reported as p=) represent the probability that the mean will be larger than 7.6 or smaller than 6.4. One of the outputs uses a sample size of 4 and one uses a sample size of 9. a. Which is which, and how do you know? b. These graphs are made so they spread out to occupy the room on the face of the calculator. If they had the same horizontal axis, one would be taller and narrower than the other. Which one would that be, and why?
3. Maryland has one of the highest per capita income in the US, with an average income of $75,847. Suppose the standard deviation is $32,000 and the distribution is right-skewed. A random sample of 100 Maryland residents is taken. a. Is the sample size large enough to use the CLT for means? Explain. b. What would be the mean and standard error for the sampling distribution? c. What is the probability that the sample mean will be more than $3200 away from the population mean? The t -Statistic Hypothesis tests and confidence intervals for estimating and testing the mean are based on a statistic called the t -statistic: Since the population standard deviation is almost always unknown , we divide by an _______________ of the _________________________, using the sample standard deviation s instead of σ . The t -Distribution The t -statistic does not follow the Normal distribution, because the denominator ______________________ with every sample size. The t -statistic is more variable than the z -statistic, whose denominator is always the _____________. If the three conditions for using the Central Limit Theorem hold, the t -statistic follows a distribution called the t -distribution . The __________________________________ (_______) represent how many values involved in a calculation have the freedom to vary. The t -Distribution is Symmetric and “bell-shaped” Has thicker tails than the Normal distribution Shape depends on the degrees of freedom ( df ) If df is small, the tails are thick; as d f increases, tails get thinner If the three conditions for using the Central Limit Theorem hold, the t-statistic follows a distribution called the t -distribution . When small sample sizes were used to make inferences about the mean , even if the population was Normal, the Normal distribution just didn’t fit the results that well. The t -distribution turned out to be a better model than the Normal for the sampling distribution of x when _______________________ ___________________________________________.
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