Copy of Chapter 8 Notes Outline - Statistical Inference Confidence Intervals PDF

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Chapter 8 Statistical Inferences: Confidence Intervals Note: As you view the lecture presentation, it is very helpful to have the notes tab showing to assist in filling in the blanks.

What’s It All About (Slide 2) List of Sample Statistics given in slide s  ample mean x  , sample proportion p , standard deviation s List of Population Parameters given in slide p  opulation mean μ , population proportion p, population standard deviation Inference methods help us to p  redict how close a sample statistic falls to the population parameter. We can make decisions and predictions about populations even if we have data for relatively few subjects from the population.

8.1 point and interval estimates of population parameters Definitions (Slide 3): Population parameters have two types of estimates, a point estimate and an interval estimate. A point estimate is a single number that is our best guess for the parameter. Most often this value comes from the sample statistic. An interval estimate is an i nterval of numbers within which the parameter value is believed to fall.

(Slide 4) A point estimate alone is not sufficient as the probability of it being exactly equal to the population parameter is not likely. (slide 5) Confidence Interval – A confidence interval is an interval containing the most believable values of a parameter. The probability that this method produces an interval that contains the parameter is called the confidence level. This is a number chosen to be close to 1, most commonly 0.95. The value of 0.95 would represent a 95% confidence interval with 95% being the confidence level.

8.2 Constructing a confidence interval to estimate a population proportion (slide 7) A few important notes: ● We symbolize the population proportion by ______. The point estimate of the population proportion is the __________________________ and we symbolize the sample proportion by __________________________. ● The formula _________________________________________________ becomes p ± 1.96(standard deviation) for a 95% confidence interval. ● The formula for the standard deviation uses p, since we do not know p, _____________ __________________________________________. (Slide 8)

(Slide 10) This procedure applies to large random samples. In practice, large means that you should have at least __________________________ and at least __________________________ for the binary outcome. Expressed as a formula, np  ≥ 15 and n ( 1-p  ) ≥ 15.

(Slide 12) Above are the confidence intervals found for the A Husband Choosing Not to Have Children example. We see that the 99% confidence interval is ___________, it is from (0.56, 0.66). The 95% confidence interval is _______________, it is from (0.57, 0.65). The 99% interval is what we call ______ _________. It has a _______________________________________. Having a greater margin of error is the sacrifice for gaining greater assurance (99% rather than 95%) of correctly inferring where p  falls. In conclusion, what we see from the slide is: 1. 2. Summary: Effects of confidence level and sample size on margin of error (slide 13) The margin of error for a confidence interval, and thus the length of the confidence interval: ● Increases as the ____________________________________ ● Decreases as the ___________________________________

What does it mean to say we have 95% confidence? (Slide 14) In the long run about 95% of the intervals would give _________________________, containing the population proportion. On the other hand, in the long run, 5% of the time the confidence interval ______ _________________________________________________________. StatCrunch Example Exit poll predictions A national television network takes an exit poll of 1400 voters after each has cast a vote in a state gubernatorial election. Of them, 660 say they voted for the Democratic candidate and 740 say they voted for the Republican candidate. We will designate the variable of interest as voting democratic, therefore we have 660 successes for this problem. 1.Find the point estimate for the population proportion, p. 2.Find a 95% confidence interval for the proportion of voters that voted democratic. State this conclusion in correction conclusion sentence form.

3.Based on this 95% confidence interval, would you predict a winner?

4.Find the 99% confidence interval.

5.Would you predict a winner based on this 99% confidence interval?

Confidence Interval For a Proportion Example A survey asked, "During the last year, did anyone take something from you by using force − such as a stickup, mugging, or threat?" Of 981subjects,  17 answered yes and 964 answered no. a. Find the point estimate of the proportion of the population who were victims. b. Find the standard error of this estimate. c. Find the margin of error for a 95% confidence interval. Explain what the margin of error represents. d. Construct the 95% confidence interval for the population proportion. Can you conclude that fewer than 10% of all adults were victims?

8.3 Constructing a Confidence Interval to Estimate a Population Mean The sample mean x is the ______________________________of the population mean μ .

Confidence interval = Point estimate ± margin of error = x  ± t· s/ √n Properties of the t Distribution (Slide 4) •The t distribution is ______________________ and symmetric about _____. •The probabilities depend on the ________________________________, df . The t  distribution has a slightly different shape for each distinct value of df , and different t-scores  apply for each df value. •The t distribution has ___________________ and has _______________________________ than the standard normal distribution. The larger the df  value, however, the ________________________ to the standard normal. When df  is about _____________________, the two distributions are nearly identical. •A t-score multiplied by the standard error gives the ___________________________ for a confidence interval for the mean.

(slide 6)Fortunately, the confidence interval using the t distribution is _____________ meaning that it performs adequately even when the assumptions are _____________________________. The most important case when the t  distribution method does not work well is when the data contain ________________________________. 3 Example Problems - Take notes on the answers and the use of StatCrunch. (Slide 7) Females’ ideal number of children The 2008 GSS asked, “What do you think is the ideal number of children for a family to have?” The 678 females who responded had a median of 2, mean of 3.22 and a standard deviation of 1.99. 1.What is the point estimate of the population mean? 2.What is the standard error (se) of the sample mean? 3.Find the 95% confidence interval for the population mean.

4.Is it plausible that the population mean μ = 2? Explain. 5.Find the margin of error. 6.Looking at the interval, where do you find the point estimate. 7.Find the 99% confidence interval for the population mean.

8.What effect did increasing the confidence level have on the length of the confidence interval? (Slide 8) High efficiency washing machines use less water and less detergent than standard washing machines. Everyone varies in the amount of detergent that they put into a washing machine. Some people might put in a little bit more; while others might use a little bit less. How many loads can you complete with a full 150 ounce bottle of leading brand laundry detergent on average? The sample data is below. 87 88 103 98 94 99 89 91 87 91 Find the 90% confidence interval for the mean number of loads. What data was collected in the study? Is it qualitative or quantitative? Do you use t stat or proportion stat in StatCrunch? State the CI inclusion here:

(Slide 8) In 2010, the GSS asked participants their opinions about spanking. They asked the following question, “Do you agree, that it is sometimes necessary to discipline a child with a good, hard spanking?” Out of 1416 respondents, 976 said that they agreed that it was sometimes necessary. Find the 95% confidence interval for this parameter. What data was collected in the study? Is it qualitative or quantitative? Do you use t stat or proportion stat in StatCrunch? State the CI inclusion here:

8.4 Choosing the Sample Size for a Study Formula for the sample size for a population proportion (Slide 9) n=

This is the sample size formula to use when the problem is about ______________________ data. We see that since we have not taken the sample yet, we do not know the value of p  so we can do one of two things, we can use a _______________ for what we believe p  might be approximately or we can be __________________ and use a p  equal to ___________. Using p  = 0.50 will result in a ______________________________ than if we have a guess of what p  might be. Formula for the sample size for a population mean (slide 10) n= Use this formula when the problem is about _______________________ data. To use this formula, you can use an ________________________ for the value of the population standard deviation. For example, you can use the _________________________________________ __________________________________ already conducted. (Slide 11) We’ve looked at two factors that play a role in determining a study’s sample size. ❏

❏

Other factors also play a role. ❏ ❏

Confidence Level

z or t

Example sample size problems, take notes on the solutions of these problems. (Slide 12)Abstainers The Harvard study estimated that 19% of college students abstain from drinking alcohol. To estimate this proportion in your school, how large of a random sample would you need to estimate it to within 0.05 with probability of 0.95, if before conducting the study A. You are unwilling to predict the proportion value at your school. B. You use the Harvard study as a guideline.

(Slide 13) Suppose the financial aid office wants to estimate the mean semester textbook cost within $25 of the true value with 99% confidence. How many students should be sampled? Previous data shows σ  is about $85.

Confidence Interval Basics presentation Quick Vocab Review - Slide 2 ________________________ – this is a measure calculated from data collected from the entire population. Population mean, ____, Population proportion, ____, and Population standard deviation, ____. _____________________ – this is a measure calculated from the sample data collected from a population. Sample ___________, x , sample ________________, p, and sample ________________________, s.

Confidence Interval Diagram - Slide 5

CI Conclusion - Slide 6 Every time you are asked to find a confidence interval, you need to state the interval in a complete sentence, with the following ____________. You need to state the ____________________________, state specifically ___________ the interval is for, in this case the mean height of all adult women in Ohio, and then ________________________________________. CI Length and Margin of Error - Slide 7 We can calculate the ____________ of the confidence interval as the ___________________________ _________________________ which in this case would be 65.6 – 64.4 = 1.2 The __________________ is found by taking ________ of the ____________ of the interval. In this case it would be length of interval ÷ 2 = 1.2 ÷ 2 = 0.6 Types of Data - Quantitative or Qualitative - Slides 8 and 10 Quantitative data – this is _____________________ such as heights, weights, temperatures, salaries, scores, etc. You may see the ________________________. You will see words such as __________ and maybe _______________________________. Always ask yourself, what was the data that was collected from each member of the sample? Was it a number? For quantitative data we are wanting to find the t test statistic so you will use t Stats in StatCrunch. Note, you will ________________________________________________________. Qualitative data – this is _________________________, such as the answer to a yes or no question. For example, do you wear a seatbelt, do you support the current president, do you believe in ghosts? Again, ALWAYS ask yourself what was the data that was collected from each member of the sample? Was it a word? You will see words like _________________________________________________ in the problem. You will be given the number of _________________ or will be able to find the number of successes. For qualitative data we are wanting to find a z test statistic and in ___________________ you would use __________________________. Precision - Slides 11 and 12 Precision is how well you have “pinned down” the value of the population parameter. A long interval would have _______ precision, you have not pinned down the value of the population parameter very well. There is a large range of values in which the population parameter could lie. A short interval would have _______ precision, you have pinned down the value of the population parameter and it lies in a small range of values. To be more precise, which means having a shorter interval, you can do one of two things: 1. _______________________________________________. 2. The second way to gain more precision is to ______________________________________.

Dart Board Example

A larger ____________________________ requires a ________________ confidence interval, you want to be very sure the population parameter lies in the interval so the interval needs to be long. However a ____________ level of confidence leads to a shorter _____________________________, with less confidence, you are okay with the parameter not always being in the interval so a shorter interval will do....