10.1 Populations & Surveys

Transcription

1 10.1 Populations & Surveys STATISTICS: the art and science of gathering, analyzing, and making predictions from numerical information (data) obtained in an experiment. A company plans to open a new restaurant in a city. To study the preferences of people in the city, the company conducts a survey of 400 city residents. SURVEYS: POPULATION SAMPLE *Statisticians use samples instead of entire populations because a) it is often impossible to obtain data on an entire population b) sampling is less expensive because collecting takes less time and effort An Unbiased Sample A Biased Sample 96 students were asked whether they prefer playing tennis or softball. 72 students preferred playing softball. Statistics can be a valuable tool to society, however it can also be misused to take advantage.

2 Identify potentially biased samples. Tell whether the sample chosen is likely to be biased. Explain. 1. To choose the theme of an allschool dance, you survey all of the members of your drama club. 2. To study support for a proposed new cell phone tower, all the households within 200 yards of the proposed site are surveyed. 3. To find out if a new Web site is easy to use, the site s designer asks five people in the office to try it. Each graph is misleading to the audience. Explain why Explain why the survey question may be biased. Then rewrite the question to remove the bias. 6. Would you like to see healthy food served in our cafeteria? 7. Do you follow dentists recommendations and floss daily? MAKING PREDICTIONS: Samples that are carefully chosen to avoid giving biased results allow you to make predictions about a population. 8. You want to find out if students at your school favor a proposed change in the school colors. Of the total of 960 students, you survey 40 who are chosen in an unbiased way. You find that 12 of the 40 favor the change. Predict how many students at your school favor the change.

3 More likely to be a BIASED sample More likely to be an UNBIASED sample Sampling Techniques: 10.2 Samples and Margin of Error RANDOM SAMPLE EXAMPLE: A $50 gift certificate is given away at the Annual Bankers Convention. Tickets are placed in a bin, and the tickets are mixed up. Then the winning ticket is selected by a blindfolded person. (Every ticket has an equal chance of being selected) SYSTEMATIC SAMPLE EXAMPLE: Every 20 th soup can coming off an assembly line is checked for defects. CONVENIENCE SAMPLE EXAMPLE: The first 50 people entering a zoo are asked if they support an increase in taxes to support a zoo expansion. SELF-SELECTED SAMPLE EXAMPLE: A university newspaper runs an ad asking for volunteers to participate in a study that focuses on the impact of Greek life on the university campus.

4 Classify each sample as random, systematic, convenience, or self-selected. Explain your choice and tell whether this is likely to be a biased or unbiased sample. 1. Employees who go to a company health fair are given survey cards that ask for opinions about new healthrelated programs. Those who fill in and return a card are given a coupon for a free Healthy Habits lunch at the company cafeteria. 2. Every company employee whose employee number ends in a 3 is surveyed. Choose a RANDOM sample. Describe two ways you can choose a random sample of 30 students to survey. 3. You plan to conduct a survey about the hobbies of students in your grade. You have a list of the names of all 240 students in your grade. Even a random sample can give a biased result just by chance! MARGIN OF ERROR FORMULA: In a random survey of 2025 workers in a city, 47.5% say that it takes them more than 30 minutes to get from home to work every day. 4. What is the margin of error for the survey? 5. Give an interval that is likely to contain the actual percent of all the city s workers who take more than 30 minutes to get to work.

5 You read that a survey of a random sample of a state s voters indicated that between 67.5% and 72.5% of the voters support a new law. 6. What is the margin of error for the sample? 7. Use the margin of error to find the size of the sample. A national chain of restaurants is studying the effects of a new company policy in its restaurants. 8. Suppose two restaurant managers from every state are surveyed. Classify the sample as convenience, selfselected, systematic, or random. 9. The table results shown are from a random sample of 100 managers. What is the margin of error for the sample? 10. Given an interval that is likely to contain the actual percent of the chain s restaurant managers who would say that the effect of the change is positive.

6 DISPERSION/POSITION CENTRAL TENDENCY MEASURES OF DISPERSION/POSITION MEASURES OF CENTRAL TENDENCY MEAN (x ) : 10.3 Measures of Central Tendency & Dispersion MEDIAN: (Second Quartile) MODE: RANGE: LOWER QUARTILE (LQ): (First Quartile) UPPER QUARTILE (UQ): (Third Quartile) INTERQUARTILE RANGE (IQR): EXAMPLE Find the mean, median, mode(s), and range of these 14 quiz scores: 7, 16, 17, 19, 20, 20, 21, 22, 23, 24, 24, 24, 25, 25 MEAN MEDIAN MODE RANGE LQ UQ IQR

7 COMPARE DATA: Use the information below: 1. Compare the mean, median, and mode of the set of original prices and the set of discounted prices. What do you observe? 2. Compare the ranges and the differences between the upper and lower quartiles for the original prices and the discounted prices. What do you observe? Box-and-Whisker Plot /Box Plot An OUTLIER is a number in the data set that doesn t fit with the rest of the data. On a box-and-whisker plot, if an outlier(s) exist, plot a SEPARATE point to the left or right of the whiskers. Smallest # Biggest # MEDIAN A box-and-whisker plot is split into four sections. Each section represents 25% of the data. If that section is longer, it means the numbers in that 25% are spread out. Remember that sometimes numbers are repeated in a set of data. Lower Quartile Upper Quartile

8 Make a Box-and-Whisker Plot of the following data: 13, 15, 11, 10, 20, 18, 8, 14, 17, 4, 15, Find the median, lower quartile, upper quartile, biggest, and smallest number first. 4. Plot your values to make the Box-and-Whisker. Median: LQ: UQ: Biggest #: Smallest #: Look at the following box-and-whisker plots and answer each question. Plot A Plot B 5. What is the median of Plot A? 6. What is the upper quartile of Plot B? 7. What is the IQR of Plot B? 8. Which plot has a smaller IQR, A or B? 9. Which plot has a larger range, A or B? 10. Which plot is more spread out, A or B? 11. In plot A, the middle 50% of data is between which two numbers? 12. In plot B, what percent of the data is greater than 52?

9 Lower class limits Upper class limits 10.3 Statistical Graphs FREQUENCY DISTRIBUTION EXAMPLE 1: The number of children per family is recorded for 64 families surveyed. Construct a frequency distribution of the following data: Data: Frequency Distribution: Number of Children Number of Families TOTAL = Often, data are grouped in classes to provide information about the distribution that would be difficult to observe if the data were ungrouped. Certain rules should apply when grouping data in classes: 1. The classes should be the same width. 2. The classes should not overlap. 3. Each piece of data should belong to only one class. 4. A frequency distribution should be constructed with only 5-12 classes. Let s consider a set of observed values that go from a low of 0 to a high of 26. That is a wide range of values from 0 to 26 so we can break these values into classes. Let s make the first class go from 0 through 4. CLASSES We say that the class width is 5 because each class has 5 integers that belong to it (0, 1, 2, 3, 4 belong to 0-4).

10 EXAMPLE 2: The following set of data represents the family income (in thousands of dollars, rounded to the nearest hundred) of 15 randomly selected families. Construct a frequency distribution with a first class of Income # Families A histogram is a graph with observed values on its horizontal scale and frequencies on its vertical scale. We use our frequency distribution to make this type of graph. HISTOGRAM EXAMPLE 3: Construct a histogram of the frequency distribution from EXAMPLE 1 below.

11 Frequency polygons are line graphs with scales the same as those of the histogram. Instead of making a bar, just put a point at the corresponding frequency and then connect each point with straight-line segments. Always put in two additional marks on the horizontal axis at the lower and upper end. FREQUENCY POLYGON HISTOGRAM & FREQUENCY POLYGON on the same graph EXAMPLE 4: Construct a frequency polygon using the frequency distribution showing the one-way commuting distances for 70 workers below. Distance (miles) Number of Workers

12 A stem-and-leaf display is a tool that organizes and groups data while allowing us to see the actual values that make up the data. STEM-and- LEAF DISPLAYS KEY: 8 1=81 years old EXAMPLE 5: Construct a stem-and-leaf plot using the data given below showing math scores on a quiz out of 50 points. Don t forget the KEY! 35, 36, 38, 40, 42, 42, 44, 45, 45, 47, 48, 49, 50, 50, 50 These graphs are often used to compare parts of one or more components of the whole to the actual whole. The total circle represents 100% so all of the percentages should add to 100%. Remember that a circle has 360 degrees total if you need to figure out about how many degrees you need for each component. CIRCLE GRAPHS/PIE CHARTS