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by: Chris Waldron

STATS 111 STAT 1111

Chris Waldron

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Kevin Beard
Class Notes
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This 9 page Class Notes was uploaded by Chris Waldron on Tuesday February 9, 2016. The Class Notes belongs to STAT 1111 at University of Vermont taught by Kevin Beard in Winter 2016. Since its upload, it has received 78 views. For similar materials see Statistics in Statistics at University of Vermont.


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Date Created: 02/09/16
Proportion The proportion in some category is found by Proportions are also called relative frequencies, and we can display them in a relative frequency  table. The proportions in a relative frequency table will add to 1 (or approximately 1 if there is  some round­off error). Relative frequencies allow us to make comparisons without referring to  the sample size. One Categorical Variable Of the n=2625 people who responded to the survey, 735 agree with the statement that there is  only one true love for each person, while 1812 disagree and 78 say they don't know.  Table 2.1 displays these results in a frequency table, which gives the counts in each category of a categorical variable. Table 2.1   A frequency table: Is there one true love for each person? Response Frequency Agree  735 Disagree 1812 Don't know   78 Total 2625 What proportion of the responders agree with the statement that we all have exactly one true  love? We have Visualizing the Data in One Categorical Variable Figure 2.1(a) displays a bar chart of the results in Table 2.1. The vertical axis gives the  frequency (or count), and a bar of the appropriate height is shown for each category. Notice that  if we used relative frequencies or percentages instead of frequencies, the bar chart would be  identical except for the scale on the vertical axis. The categories can be displayed in any order on the horizontal axis. Another way to display proportions for a categorical variable, common in the popular media, is with a pie chart, as in Figure 2.1(b), in which the proportions correspond to the areas of sectors of a circle. Notation for a Proportion As we saw in Chapter 1, it is important to distinguish between a population and a sample. For this reason, we often use different notation to indicate whether a quantity such as a proportion comes from a sample or an entire population. Notation for a Proportion The proportion for a sample is denoted and read “p-hat”. The proportion for a population is denoted p. Two Categorical Variables: Two-Way Tables Does the proportion of people who agree that there is exactly one true love for each person differ between males and females? Does it differ based on the education level of the responders? Both questions are asking about a relationship between two categorical variables. We investigate the question about gender here and investigate the effect of education level in Exercise 2.18. To investigate a possible relationship between two categorical variables we use a two-way table. In a two-way table, we add a second dimension to a frequency table to account for the second categorical variable. Table 2.3 shows a two-way table for the responses to the question of one true love by gender. Two-Way Table A two-way table is used to show the relationship between two categorical variables. The categories for one variable are listed down the side (rows) and the categories for the second variable are listed across the top (columns). Each cell of the table contains the count of the number of data cases that are in both the row and column categories. Table 2.3 Two-way table: Is there one true love for each person? Male Female Agree 372  363 Disagree 807 1005 Don't know  34   44 It is often helpful to also include the totals (both for rows and columns) in the margins of a two-way table, as in Table 2.4. Notice the column labeled “Total” corresponds exactly to the frequency table in Table2.1. Table 2.4 Two-way table with row and column totals Male Female Total Agree 372  363  735 Disagree 807 1005 1812 Don't know  34   44   78 Total 1213 1412 2625 So, are men or women more hopelessly romantic? The two-way table can help us decide. Use Table 2.4 to answer the following questions. (a)  What proportion of females agree? (b)  What proportion of the people who agree are female? (c)  What proportion of males agree? (d)  Are females or males more likely to believe in one true love? (e)  What proportion of survey responders are female? (a) To determine what proportion of females agree, we are interested only in the females, so  we use only that column. We divide the number of females who agree (363) by the total  number of females (1412): (b)  To determine what proportion of the people who agree are female, we are interested only in  the people who agree, so we use only that row. We have Notice that the answers for parts (a) and (b) are NOT the same! The proportion in part (a) is  probably more useful. More females than males happened to be included in the survey, and this  affects the proportion in part (b), but not in part (a). (c)  To determine what proportion of males agree, we have (d)  We see in part (c) that 31% of the males in the survey agree that there is one true love for  each person while we see in part (a) that only 26% of the females agree with that statement. In  this sample, males are more likely than females to believe in one true love. (e)  To determine what proportion of all the survey responders are female, we use the total row.  We have We see that 54% of the survey respondents are female and the other 46% are male. Example 2.6   In the StudentSurvey dataset, students are asked which award they would prefer to win: an Academy Award, a Nobel Prize, or an Olympic gold medal. The data show that 20 of the 31 students who prefer an Academy Award are female, 76 of the 149 students who prefer a  Nobel Prize are female, and 73 of the 182 who prefer an Olympic gold medal are female. (a) Create a two­way table for these variables. (b) Which award is the most popular with these students? What proportion of all  students selected this award? (a)  The relevant variables are gender and which award is preferred. Table 2.5 shows a two­way table  with three columns for award and two rows for gender. It doesn't matter which variable corresponds to  rows and which to columns, but we need to be sure that all categories are listed for each variable. The  numbers given in the problem are shown in bold, and the rest of the numbers can be calculated  accordingly. Table 2.5   Two­way table of gender and preferred award   Academy Nobel Olympic Total Female 20  76  73 169 Male 11  73 109 193 Total 31 149 182 362 (b)  More students selected an Olympic gold medal than either of the others, so that award is the most  popular. We have We see that 50.3%, or about half, of the students prefer an Olympic gold medal to the other options. Example 2.7   In Example 2.6, we see that the most popular award is the Olympic gold medal. Is preferring an  Olympic gold medal associated with gender? Use Table 2.5 to determine the difference between  the proportion of males who prefer an Olympic gold medal and the proportion of females who  prefer an Olympic gold medal. Since the data come from a sample, we use the notation   for a proportion, and since we are  comparing two different sample proportions, we can use subscripts M and F for males and  females, respectively. We compute the proportion of males who prefer an Olympic gold  medal,  , and the proportion of females who prefer an Olympic gold medal,  , The difference in proportions is Males in the sample are much more likely to prefer an Olympic gold medal, so there appears to  be an association between gender and preferring an Olympic gold medal. As in Example 2.7, we often use subscripts to denote specific sample proportions and take the  difference between two proportions. Computing a difference in proportions is a useful measure  of association between two categorical variables, and in later chapters we develop methods to  determine if this association is likely to be present in the entire population. Visualizing a Relationship between Two Categorical Variables There are several different types of graphs to use to visualize a relationship between two  categorical variables. One is a segmented bar chart, shown in Figure 2.2(a), which gives a  graphical display of the results in Table 2.5. In a segmented bar chart, the height of each bar  represents the number of students selecting each award, while the color (in this case, red for  females and green for males) indicates how many of the students who preferred each type were  male and how many were female.


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