# Chapter 3 Generalization: How broadly do the results apply?

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Chapter 3 Generalization: How broadly do the results apply?

Generalization So far we’ve studied significance and estimation.
Once we make a conclusion from a test of significance or construct a confidence interval, how broadly do these apply or to what population can I generalize these results? This generalization is the topic for this section.

Generalization Sometimes this generalization is difficult and sometimes it is not. Generalizing to a larger population is valid only when the sample is representative. Unfortunately, biased sampling methods are common.

Section 3.1 Introduction to sampling from a finite population

Notation Check Statistics Parameters 𝑥 (x-bar) Sample Average or Mean
𝑝 (p-hat) Sample Proportion Parameters 𝜇 (mu) Population Average or Mean 𝜋 (pi) Population Proportion Statistics summarize a sample and parameters summarize a population

Sampling Hope College students
Suppose we want to know the proportion of Hope students that watched the Super Bowl. Or the average number of traffic tickets Hope students have received. The population of interest is all Hope students. A census will get this information from all Hope students. What if you don’t have time/money to interview all students?

Sampling We can take a sample of Hope students and find the proportion of those in our sample that watched the Super Bowl or mean number of traffic tickets they have received. Using these statistics we can make inferences to the parameters. How well will these statistics represent our parameters of interest? The key to this question is how the sample is selected from the population.

Random Sampling Getting a random sample is key to making a good inference. This can be tough; we don’t live in a random world. For example, the people you see on a daily basis can be very different from the people others near to you see on a daily basis. When samples are not random or representative their results can be misleading.

Biased Sampling

ESPN Top 10: What is college basketball's fiercest rivalry?
Connecticut vs. Tennessee (Women) Duke vs. North Carolina Hope vs. Calvin Illinois vs. Missouri Indiana vs. Purdue Louisville vs. Kentucky Penn vs. Princeton Philadelphia's Big 5 Oklahoma vs. Oklahoma State Xavier vs. Cincinnati

ESPN Top 10: What is college basketball's fiercest rivalry?
75.1% Hope vs. Calvin 9.3% Duke vs. North Carolina 5.4% Indiana vs. Purdue 5.2% Philadelphia's Big % Penn vs. Princeton 1.5% Oklahoma vs. Oklahoma State 0.7% Louisville vs. Kentucky 0.6% Connecticut vs. Tennessee (Women) 0.3% Illinois vs. Missouri 0.3% Xavier vs. Cincinnati Total Votes: 46,084

2012 State ACT Results New York ranked 6th with an average of 23.3.
Michigan ranked 45th with an average of 20.1.

2011 State SAT Results ??? MI NY ACT 100% 29% SAT 4% 90%
New York ranked 45th with an average of 1466. Michigan ranked 6th with an average of 1762. ??? MI NY ACT % % SAT % %

Random Sample To have a random sample, you can’t have people self-select themselves into the sample. (Basketball poll) You can’t choose a convenient sample that is clearly not representative of the population. (ACT vs. SAT)

Random Sample A simple random sample is the easiest way to ensure that your sample is unbiased. A sampling method is biased if statistics from samples consistently over or under- estimate the population parameter.

Simple Random Sample A simple random sample is like drawing names out of a hat. Technically, a simple random sample is a way of randomly selecting members of a population so that every sample of a certain size from a population has the same chance of being chosen.

Sampling Every simple random sample gives us different values for the statistics. There is variability from sample to sample (sampling variability). If we take repeated simple random samples of Hope students, each sample will consist of different students. We will get different means or proportions each time we do this. However …

Sampling The sample means or proportions will center around the population mean or proportion if the sampling method is unbiased (like a simple random sample). Our sampling variability will decrease when we take larger and larger sample sizes.

Exploration 3.1A: Sampling Words
We need to sample from a population of interest if it is very large or is difficult to measure every single member of the population. If we were interested in High School GPA for Hope students we would not need to sample. The registrar’s office has all that information. If we were interested in something that has not already been collected, we might want to sample.

Exploration 3.1A: Sampling Words
That being said, in this activity we will be using the words in the Gettysburg Address as our population. There are fewer than 300 in this speech and we could easily look at the entire speech to find out average word length, proportion of words that contain an e, etc. We will be sampling from this speech not to get information from the population, but to help us learn some things about sampling.

Only picture of Lincoln at Gettysburg (Edward Everett spoke for over two hours. Lincoln followed with his two-minute speech.)

Exploration 3.1A Select what you think is a representative sample of 10 words from the Gettysburg (pg 3-10). Record your words in table in question 2. Make dotplots of both average length and proportion containing e on the board. Only work through question 22. HW: Exercises and 3.1.4

Review of Section 3.1 A sampling method is biased if statistics from samples consistently over or under- estimate the population parameter. A simple random sample is the easiest way to insure that your sample is unbiased. Therefore, if we have a simple random sample, we can infer our results to the population from which is was drawn.

Review of Section 3.1 We saw biased and unbiased sampling in the Gettysburg Address exploration. We also saw that: When we increase sample size, the variability of our sampling distribution decreases. This variability can be predicted. Changing the population size has no effect on variability.

Population distribution of word lengths
Distribution of average word length from samples of size 20

Section 3.2: Inference for a Single Quantitative Variable
Using methods similar to what we did in the last section, we will see how a null distribution for a single quantitative variable can be obtained and even predicted.

Example 3.2: Estimating Elapsed Time

Estimating Time Does it ever seem that time drags or flies by?
Students in a stats class (for their project 2) collected data on students’ perception of time Subjects were told that they’d listen to music and asked questions when it was over. Played 10 seconds of the Jackson 5’s “ABC” and asked how long they thought it lasted Can students accurately estimate the length?

Hypotheses Null Hypothesis: People will accurately estimate the length of a 10 second-song snippet, on average. (μ = 10 seconds) Alternative Hypothesis: People will not accurately estimate the length of a 10 second-song snippet, on average. (μ ≠ 10 seconds)

Estimating Time A convenience sample of 48 students on campus were subjects and song length estimates were recorded. The average estimate was sec and the standard deviation was 6.50 sec.

Skewed, mean, median The distribution obtained is not symmetric, but is right skewed. When data are skewed right, the mean gets pulled out to the right while the median is more resistant to this.

Mean v Median The mean is 13.71 and the median is 12.
How would these numbers change if on of the people that gave an answer of 30 seconds actually said 300 seconds? The standard deviation is 6.5 sec. Is it resistant to outliers?

Population? One way to develop a null distribution is to draw samples from some population that we think our population of time estimates might look like under a true null. Under the null the mean is 10 sec. We might assume the population is skewed and has a standard deviation similar to what we found.

Simulation-based Inference
We have a possible population data set similar to what we need. Let’s go and get that data. Then go to the One Mean applet and develop a null distribution. Find out where our actual mean of sec is located. And finally see how a t-distribution could predict all this.

T-distribution The t-distribution is very similar to a normal distribution, but with slightly “heavier” tails. The t-statistic is the standardized statistic we use with a single quantitative variable and can be found using the formula: 𝑡= 𝑥 −𝜇 𝑠 𝑛

Validity Conditions The theory-based test for a single mean requires either: The sample size is at least 20. If the sample size is less than 20 the sample distribution is not skewed. Let’s use the theory-based applet to run this test and find a confidence interval. (We first need to get the data.)

Estimating Time Formulate Conclusions.
Based on our small p-value, we can conclude that people don’t accurately estimate the length of a 10-second song snippet and in fact they overestimate it. To what larger population can we make our inference?

Estimating Time We are 95% confident that the average estimate of a 10 second song is between and seconds.

Exploration 3.2: Sleepless Nights?
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