AP Stats – Chapter 10 - Weebly



AP Stats – Chapter 10

Section 10.1: Confidence Intervals: The Basics

How is this chapter different from chapter 9?

What is statistical inference? What assumptions are we making when we perform inference techniques?

What is a point estimate?

Example 1: IQ Tests

Suppose we give IQ tests to 50 AHS seniors and the mean score is 112. What can we say about [pic], the mean IQ score of ALL AHS seniors?

a) Is [pic]? Should it be close?

b) What’s the standard deviation of the sampling distribution? (Let’s assume [pic])

c) What should the sampling distribution look like?

d) We can create an interval for the true value of [pic]. In 95% of all samples, xbar will lie within 2 standard deviations of the true mean [pic]. Let’s use this information to create an interval for the true mean (remember, this is just an estimate).

What is a confidence interval?

What is the confidence level?

What is the margin of error?

Why do we include the margin of error?

Example 2: How old is my husband?

a) Write down an interval that you think captures the true age of Mrs. H’s husband. How confident are you that you’ve captured the true age?

b) What if I told you that if you got it wrong, I would cut off your hand (not really…but for the sake of the problem just go with it ( )! Do you want to adjust your interval? How might you want to adjust it? How confident are you that you captured the true age?

c) Now, what if I told you that if you got it wrong I would cut off your hand and both feet (again, just go with it ( )! Do you want to make another adjustment? How confident are you now?

d) What should be happening to your intervals? What’s happening to your confidence level?

How do you interpret a confidence level? In other words, what does it mean to be 90% confident?

How do you interpret a confidence interval?

Example 3: According to , on August 13, 2010, the 95% confidence interval for the true proportion of Americans who approved of the job Barack Obama was doing as president was 0.44 [pic] 0.03. Interpret the confidence interval and the confidence level.

What three conditions need to be met to calculate a confidence interval?

What is the formula for calculating a confidence interval? Is this formula included on the formula sheet?

What are the three most common critical values?

What is the four-step process for calculating a confidence interval?

Example 4: Suppose we want to estimate the mean SAT math score for all seniors in GA. A simple random sample of 500 students gives a mean of 461. Assume [pic].

a) Construct and interpret a 99% confidence interval for the true mean SAT math score for all seniors in GA.

b) Construct and interpret an 85% confidence interval for the true mean SAT math score for all seniors in GA.

How can I do this on my calculator?

Ideally, we want a high confidence level and a small margin of error. How can we reduce the margin of error in a confidence interval?

Example 5: What sample size do we need if we want to be 95% confident and have a margin of error of 1? Assume [pic].

Example 6: Administrators at your school want to estimate how much time students spend on homework, on average, during a typical week. They want to estimate[pic] at the 90% confidence level with a margin of error of at most 30 minutes. A pilot study indicated that the standard deviation of time spent on homework per week is about 154 minutes. How many students need to be surveyed to estimate the mean number of minutes spent on homework per week with 90% confidence and a margin of error of at most 30 minutes?

Section 10.3: Estimating a Population Proportion

Characteristics of a sampling distribution of sample proportions:

In practice, we don’t actually know p, so we replace it with __________. That also changes the above characteristics too:

What three conditions do you have to check when creating a confidence interval for a proportion?

What is the formula for calculating a confidence interval for a proportion?

Example 7: A Harvard School of Public Health survey found that 2486 of a sample of 10, 904 college undergraduates said they had engaged in frequent binge drinking. We will act as if the sample were an SRS. Construct and interpret a 99% confidence interval for the true proportion, p, of all undergraduates who admit to frequent binge drinking.

How do I do this on my calculator?

Example 8: You work for a company who has received complaints about its customer service. They intend to hire a consultant to carry out a survey of customers. Before contacting the consultant, the company president wants some ideas of the sample size that she will be required to pay for. The president wants to estimate the proportion, p, of customers who are satisfied. She decides she wants the estimate to be within 3% at a 95% confidence level. How large should the sample be to meet her criteria?

Example 9: Suppose that you wanted to estimate p = the true proportion of students at your school who have a tattoo with 98% confidence and a margin of error of no more than 0.10. How many students should you survey?

Section 10.2: Estimating a Population Mean

In the first section, the population standard deviation, [pic], was given to you. This is very unrealistic for what would actually happen in the real world. Remember, we never really know the true population parameters.

In this section we will not know the true population standard deviation. Instead we will estimate it with

_______________________________________________________.

When we use [pic] instead of [pic], the result is not a Normal distribution….it is a ___________________.

When do we use the t-distribution?

Characteristics of the t-distribution:

What is the formula for a confidence interval for a population mean? Is this formula on the formula sheet?

What are the three conditions for constructing a confidence interval for a population mean?

Example 10:

a) Suppose you wanted to construct a 90% confidence interval for the mean[pic] of a Normal population based on an SRS of size 10. What critical value t* should you use?

b) What if you wanted to construct a 99% confidence interval for [pic] using a sample of size 75?

Example 11: Environmentalists, government officials and vehicle manufacturers are all interested in studying the auto exhaust emissions produced by motor vehicles. The major pollutants in auto exhaust from gasoline engines are hydrocarbons, monoxide and nitrogen oxides (NOX). The following data set gives the NOX levels (in grams per mile) for a random sample of light duty engines of the same type.

1.28 1.17 1.16 1.08 0.6 1.32 1.24 0.71 0.49 1.38 1.2 0.78

0.95 2.2 1.78 1.83 1.26 1.73 1.31 1.8 1.15 0.97 1.12 0.72

1.31 1.45 1.22 1.32 1.47 1.44 0.51 1.49 1.33 0.86 0.57 1.79

Construct and interpret a 95% confidence interval for the mean amount of NOX emitted by light duty engines of this type.

How do I do this on my calculator?

Example 12: The principal at a large high school claims that students spend at least 10 hours per week doing homework, on average. To investigate this claim, an AP Statistics class selected a random sample of 250 students from their school and asked them how long they spent doing homework during the last week. The sample mean was 10.2 hours and the sample standard deviation was 4.2 hours.

(a) Construct and interpret a 95% confidence interval for the mean time that students at this school spent doing homework in the last week.

(b) Based on your interval in part (a), what can you conclude about the principal’s claim?

What is a paired t procedure?

What are the different ways to do a paired t procedure?

Example 13: Is caffeine dependence real? Our subjects are 11 people diagnosed as being dependent on caffeine. Each subject was barred from coffee, colas and other substances containing caffeine. Instead, they took capsules containing their normal amount of caffeine. During a different time period, they took placebo capsules. The order in which the subjects took caffeine and placebo was randomized. The data set contains data on one of several tests given to the subjects. “Depression” is the score on the Beck Depression Inventory. Higher scores show more symptoms of depression.

|subject |deprscaf |deprsplac |

|1 |5 |16 |

|2 |5 |23 |

|3 |4 |5 |

|4 |3 |7 |

|5 |8 |14 |

|6 |5 |24 |

|7 |0 |6 |

|8 |0 |3 |

|9 |2 |15 |

|10 |11 |12 |

|11 |1 |0 |

Construct a 90% confidence interval for the mean change in depression score.

Example 14: Is the express lane faster?

Libby and Kathryn decided to investigate which line was faster in the supermarket: the express lane or the regular lane. To collect their data, they randomly selected 15 times during a week, went to the same store, and bought the same item. However, one of them used the express lane and the other used a regular lane. To decide which lane each of them would use, they flipped a coin. If it was heads, Libby used the express lane and Kathryn used the regular lane. If it was tails, Libby used the regular lane and Kathryn used the express lane. They entered their randomly assigned lanes at the same time, and each recorded the time in seconds it took them to complete the transaction. Construct and interpret a 99% confidence interval for the average difference in their times.

|Time in |Time in |

|express lane |regular lane |

|(seconds) |(seconds) |

|337 |342 |

|226 |472 |

|502 |456 |

|408 |529 |

|151 |181 |

|284 |339 |

|150 |229 |

|357 |263 |

|349 |332 |

|257 |352 |

|321 |341 |

|383 |397 |

|565 |694 |

|363 |324 |

|85 |127 |

Random selection of subjects allows us to _____________________________________________

Random assignments to treatment groups allows us to _________________________________________

What is a robust inference procedure?

AP Statistics Chapter 10 – Estimating with Confidence Summary of Procedures

|Estimate a population… |Proportion |Mean with the population standard deviation |Mean without the population standard deviation |

|Appropriate Statistical Procedure | | | |

|Point estimate | | | |

|Formula for the Measure of Spread | | | |

|Condition: _____________ | | | |

|Condition: _____________ | | | |

|Condition: _____________ | | | |

|Which distribution and table are used? | | | |

|Formula for a confidence interval | | | |

|Generic Interpretation | |

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