Sindy Saintclair Monday, November 28 2021 Lesson 7 – Uniform, Binomial, Student’s T, and F-Distributions Learning Objectives and Questions Notes and Answers Understan d the properties of the uniform and binomial distributio ns The Uniform Distribution All options have an equal opportunity of being selected Looks flat and rectangular in shape if discrete, or in other words whole numbered If on a continuous variable, the same box shape would be seen but, on a histogram, there would be little variations because of what comes after the decimal place. Not very common and does not play a large role in statistics Parameters of the Continuous Uniform Distribution Mean – midpoint between the min and max Median – same as the mean Range – max minus min Standard Deviation – 30% of the range Though the bars are not labeled in this graph, suppose they are labeled 8 – 18. Since the variable is discrete, the value of 9.3 will never happen, so the probability of that outcome is 0. Each number from 8 – 18 is possible and equally as likely as any other. Each number 8 – 18 has a probability of about 0.091 of occurring. However, numbers less than 8 or greater than 18 are not possible, so their probability is 0.

A common example of a discrete uniform variable is the rolling of the single 6-sided die. Each number 1 -6 has an equal probability of occurring (1/6 or 0.167), and it is impossible to roll a 0 or a 7 or greater with a single die. A spinner with equal sized pie shape pieces is another example of a discrete uniform variable. Population Parameters of the Uniformly Distributed Continuous Variable These numbers were generated using 2 and 4 as the boundaries. The data are bucketed into buckets that are 0.1 units wide, and it gives the impression of being discrete, but you can imagine if the ‘curve’ was smoothed a bit, and infinitely many random numbers were created, it would look a lot like the continuous distribution shown above. Mean: 3 Median: 3 Standard Deviation: 0.578 Min: 2 Max: 4 Range: 2 The Binomial and Multinomial Distributions Binomial – when you have multiple trials that either end in a success or failure – only 2 outcomes such as heads or tails or life or death. - commonly used For example, if you wanted a pink poodle, then it would be considered a success and all other outcomes would be considered failures. Another example would be taking allergy medication. If

sample, dependen t, and independe nt t test in MS Excel - instead of testing against the normal distribution, it uses the student’s t-distribution, which looks a lot like the normal distribution and looks more alike as the sample size gets bigger - eventually with a large enough sample size, it may look like a normal distribution - both the z-test and the single sample t-test compare a score to the population - Another difference between normal distribution and the t distribution is that the t distribution adds another parameter such as the degrees of freedom (df) How does the distribution shape change with sample size? df = n – 1 Although different analyses have different ways to calculate df, they will all depend on n, the sample size. Example: Hobbit Movie Rating Example x̅ = 4.25; µ = 4.00  difference was so large because the n, or sample size was 10,000. That displays such a significant difference. The n is sometimes disguised in the degrees of freedom. µ = population mean s = sample standard deviation n = sample size x̅ = individual score The Student’s t-Distribution In the early 1900s, a man named William Gosset- worked at the Guinness brewery in Dublin. He was exploring how to make determinations about populations with sample sizes that could be quite small. One of the things he was looking into was chemical properties in barley when the sample size was as small as 3. Even though he didn’t invent the method, he published his findings in Biometrika . He used the pseudonym ‘student.’ This is why his work has gone under the name “Student’s t.” - a good way to determine probabilities for normally distributed populations where the population standard deviation (sigma =σ) is unknown. In order to determine probabilities, one more parameter needs to be explained. There is this thing called degrees of freedom for the t-distribution. Degrees of freedom will always

be associated with the sample size. For the t-distribution, if the sample size is n , then the degrees of freedom is ( n -1). Below is the t-distribution for 3 degrees of freedom (df): Below is the t-distribution for 15 df: I note that the t-distribution is very similar to the normal distribution. The normal distribution is overlaid on each of the above graphs of the t-distribution, in a light grey color. As the df increases, the t-distribution looks more and more like the normal distribution, to the point that at 30 or more df, the t- distribution and the normal distribution are indistinguishable. The t-distribution is useful for determining probabilities when sigma is unknown. If you have a situation where you think the population mean is mu, and you take a sample of size n from that population, you can then calculate what is often called the t-score. There are 4 variables in this equation. There is the population mean, There is the size of the sample used, n There is the sample standard deviation, s There is the sample mean, x̅ This equation looks similar to the z- score equation used in the previous lesson, and it is. In general, the main difference is that the z-score is used when sigma is known , and the t-score is used when sigma is not known.

The probability associated with these values is 0.0133, which mean that if you assume the population of widgets has a ‘warpage’ point of 305 degrees, then the probability that you would get a t = -2.92 or lower (which is why I highlighted the left tail) is 0.0133. In other words, it is not likely. It is more likely that my assumption that the average warpage begins at 305 degrees is wrong. Dependent t-Tests Paired Data Paired vs Independent Data is paired, or dependent, when they are linked together in some way. For example, if I have the same person’s data at different time points, they are linked together by the person. If, for each experimental unit in the first sample, there is a corresponding experimental unit of the second example, then the samples are probably paired. Here are some examples of paired data: - pre- and post-scores are taken before and after a new training program. The pre- and post- scores for an individual are paired. - BP is measured before and after Tx for several pts. The before and after measurements for each patient are paired. - Several adults are given 2 different exams covering the same material. The test proctor is trying to determine if the tests are essentially the same level of difficulty. The two test scores for each individual are paired. - 2 different brands of bicycle tires are being compared to see if one of them wear better than the other. Several bikes are equipped with one tire from each brand, and they are given to subjects to ride for 3 months, and the amount of wear is measured for each tire. Each individual bike will produce two measurements, one for each tire. These two measurements for each bike will be

paired. Calculating the dependent t-test by hand Dependent t-test equation It can be read as d bar minus 0 divided by the standard deviation of d bar. D bar is the mean of the difference between the first and second score for each count. The S d bar is the standard error of the differences between the first and second score. Standard Error of the Difference Equation Calculating the Standard Error of the Difference S D̅ = 59.23 / sqrt(9) = 59.23 / 3 = 19.74 Calculating the Dependent t-Test t = -5.61 – 0 / 19.74 = -0.28 The smaller the t value, the less likely it would be significant When the df of 8, and the t value 0.28 are inserted into the Student’s t Probabilities applet, the p value is 0.7866, which is higher than 0.05. This means that I must accept the null hypothesis, which means that there is no difference between cats drinking water out of the bowl and cats drinking water out of the faucet. So why do you care if the samples are paired? When comparing two samples, you are trying to see if the amount of

Case Study: Concussion Rates in Male and Female Athletes

State the Null and Alternative Hypotheses and the Level of Significance The null hypothesis is that there is no difference in the number of concussions for men and women.  H 0 : μ 1 = μ 2 The alternative hypothesis is that men and women will have different rates of concussion.  H a : μ 1 ≠ μ 2 The alpha is 0.05 level of significance. Give the relevant summary statistics I will use x bar1 to denote the mean number of concussions for men, s1 for the male standard deviation, and n1 for the male concussion sample size. For women, you will use x-bar2, s2, and n2, respectively. Test for Assumptions For an independent t test to be accurate, the data must be normally distributed for each group. I must examine the groups separately otherwise it would obscure any differences between the groups. Neither of them look normally distributed! No bell-shaped curve here. Give the Test Statistic and its Value The test statistic for a hypothesis test comparing two means with independent samples is a t. The calculation for the two sample t-test is not trivial. It can be calculated by hand, but in trying to be consistent with the goal of keeping calculations

Plug in the values: -5.61 / 59.23 = -0.95 Effect Size Small less than or equal to 0.2 Medium 0.3 – 0.5 Large greater than or equal to 0.6 a better indicator than p value of how strong the findings of a particular test are, because p value depends on sample size via the degrees of freedom, but effect size does not. You can calculate effect size using a measure called Cohen’s D. Learn about the F- Distributi on and the role it plays in ANOVAs The F Distribution For what is the F Distribution Used? - analysis of variance (ANOVAs) - Regressions – modeling - Data comparing more than 2 groups Compares one group to another, 2 or more groups More skewed than t test distribution; its shape is determined by the degrees of freedom, m and n Does not approximate the normal distribution The n and the m in the figure correspond to the two values of degrees of freedom. Please note from looking at this graph that a value for F less than 0 is impossible. The “peak” of the distribution is usually around 1, and the distribution goes on forever to the right side. Much like the normal distribution, the right side the curve never actually touches the horizontal axis, but gets closer as it gets farther out.

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