8 5 Sampling Distributions

Transcription

1 8 5 Sampling Distributions Skills we've learned 8.1 Measures of Central Tendency mean, median, mode, variance, standard deviation, expected value, box and whisker plot, interquartile range, outlier 8.2 Data Gathering population, census, sample, random sample, convenience samples, and self selected samples, underrepresented, overrepresented, bias samples, statistic, parameter 8.3 Surveys, Experiments, & Observational Studies experiment, observational study, controlled experiment, treatment group, control group, randomized comparative experiment 8.4 Significance of Experimental Results hypothesis testing, null hypothesis, z Test, 95% confidence level: If, then you can reject the null hypothesis with 95% certainty. If, then you do not have enough evidence to reject the null hypothesis. What are you trying to do? 1) Generalize: Randomly Select, using surveys or observational studies 2) Cause & Effect: Randomly Assign, using experimental designs. 8 5 Sampling Distributions 1. Estimate population means and proportions and develop margin of error from simulations involving random sampling. 2. Analyze surveys, experiments, and observational studies to judge the validity of the conclusion. Pollsters use different survey methods to accurately reflect public opinion.

2 I. Types of Samples II. Classifying a Sample 1. The campaign staff for a state politician wants to know how voters in the state feel about a number of issues. Classify each sample. A. They call every 50th person on a list of registered voters in the state. This is a systematic sample as members are chosen using a pattern. B. They randomly select 100 voters from each county to call. This is a stratified sample as the county is chosen and then voters are selected at random. C. They ask every person who comes to the next campaign rally to fill out a survey. This is a convenience sample as the people at the rally are easily accessible.

3 2. The editor of a snowboarding magazine wants to know the readers favorite places to snowboard. The latest issue of the magazine included a survey, and 238 readers completed and returned the survey. Classify the sample. This is a self selected example as readers volunteered to participate. III. Evaluating Sampling Methods A probability sample is a sample where every member of the population being sampled has a nonzero probability of being selected. Simple random samples, stratified samples, and cluster samples are all examples of probability sampling. Convenience sampling and self selected sampling are examples of non probability sampling. Most Accurate Very Accurate Not Very Accurate Census Simple random sample Stratified sample Cluster sample Convenience sample Self selected sample

4 3. A community organization has 56 teenage members, 103 adult members, and 31 senior members. The council wants to survey the members. Classify each sampling method. Which is most accurate? Which is least accurate? Explain your reasoning. Method A: simple random Method B: systematic Method C: Stratified Method A is the most accurate because every member of the population is equally likely to be in the sample. In Method C, the sample contains an equal number from each group, but the total numbers in each group differ significantly. So, adults are underrepresented and seniors are overrepresented. Method B is the least accurate because members who do not attend the cleanup have no chance of being included. 4. A small town newspaper wants to report on public opinion about the new City Hall building. Classify each sampling method. Which is most accurate? Which is least accurate? Explain your reasoning. Method A: self selected sample Method B: convenience sample Method C: cluster sample Method A is the least accurate because only people who are willing to volunteer their opinions are chosen. Method B is also inaccurate because only students and only those in the cafeteria are surveyed. Method C is the most accurate because different groups are randomly chosen and then all members of the chosen group are surveyed.

5 IV. Margin of Error The margin of error of a random sample defines an interval, centered on the sample percent, in which the population percent is most likely to lie Students at a high school will vote on a proposal to start classes later in the day. According to a survey of a random sample of students, 54% of the students agree with the proposal and 46% of the students disagree with the proposal. The survey's margin of error is ± 5%. Does the survey clearly project the outcome of the voting? Step 1 Use the margin of error to find an interval in which the actual percent of students who agree with the proposal is likely to lie. 54% ± 5% represents the interval 49% to 59%. Step 2 Use the margin of error to find an interval in which the actual percent of students who disagree with the proposal is likely to lie. 46% ± 5% represents 41% to 51%. You can conclude that between 49% and 59% of all students agree with the proposal and between 41% and 51% of all students disagree with the proposal. Because the intervals overlap, the survey does not clearly project the outcome of the voting. 5. A city is about to hold an election. According to a survey of a random sample of city voters, 42% of the voters plan to vote for Poe and 58% of the voters plan to vote for Nagel. The survey s margin of error is ±7%. Does the survey clearly project the outcome of the voting? Between 35% and 49% of all voters plan to vote for Poe and between 51% and 65% of all voters plan to vote for Nagel. Because the intervals do not overlap, the survey does clearly project the outcome of the voting.

6 6. A survey of a random sample of voters shows that 38% of voters plan to vote for Gonzalez, 31% of voters plan to vote for Chang, and 31% plan to vote for Harris. The survey has a margin of error of ±3%. Does the survey clearly project the outcome of the voting? Explain. Yes; while there is overlap between the intervals for Chang and Harris, their intervals, which are from 28% to 34%, do not overlap the interval for Gonzalez, which is 35% to 41%. 8.5 p.583 #1 15, 17, 18, 20, 23 26, 28