Ways to Measure Central Tendency



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Period _______ Date ___________________

|10.1 Confidence Intervals: The Basics |

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|What is statistical inference? | |

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|What is a point estimate? | |

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|Problem 1 – IQ Tests |

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|Suppose we give IQ tests to 50 AHS seniors and the mean score is 112. Assume [pic]. |

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|a) Let’s say [pic] is the IQ score of ALL AHS seniors. Is [pic]? |

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|b) What’s the standard deviation of the sampling distribution? |

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|c) What should the sampling distribution look like? |

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|d) We can create an interval for the true value of [pic]. In 95% of all samples, x(bar) will lie within 2 standard deviations of the true mean [pic]. |

|(recall empirical rule). Let’s use this information to create an interval for the true mean (remember, this is just an estimate). |

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|What is a confidence interval? | |

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|What is a confidence level? | |

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|What is the margin of error and why | |

|is it included? | |

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|Problem 2 – How confident are you? |

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|a) Write down what you think is the age of Mrs. Hetherington’s husband. How confident are you that you’ve captured his true age? |

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|b) Now write down an interval that you think captures my husband’s age…but if you get it wrong I’m going to give you a zero as a test grade. Do you want |

|to adjust your interval? How confident are you that you’ve captured his true age? |

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|c) Now, what if I told you that if you got it wrong you got zeroes for three test grades? Do you want to adjust your interval? How confident are you now?|

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|d) Now, what if I told you that if you got it wrong I will cut off both of your hands?. Do you want to adjust your interval? How confident are you now? |

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

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|How is a confidence level | |

|interpreted? | |

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|What does it mean to be 90% | |

|confident? | |

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|How is a confidence interval | |

|interpreted? | |

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|Problem 3 – Obama approval ratings |

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|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. |

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|What three conditions need to be met |1. |

|to calculate a confidence interval? | |

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| |3. |

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|Confidence Interval Formula | |

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|Three common critical values | | | |

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|What is the four-step process for |1. |

|calculating confidence intervals? | |

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| |4. |

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|Problem 4 – SAT Math Scores |

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|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]. |

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

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

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|How can the calculator help us find | |

|the confidence interval? | |

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|How can we reduce the margin of error| |

|in a confidence interval? | |

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|Problem 5 – Sample Size |

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

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|Problem 6 – Homework Time |

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|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? |

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|10.3 Estimating a Population Proportion |

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|Problem 7 – State Sales Tax (Chapter 9 review) |

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|A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to |

|education. Suppose that 40% of all adults in Ohio supports the increase. Find the probability that the sample proportion is between 0.38 and 0.42. |

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|What are the characteristics of a | |

|sampling distribution of a sample | |

|proportion? | |

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|Since we don’t usually know the value| |

|of p, what do we use instead? | |

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|What three conditions need to be met |1. |

|to calculate a confidence interval | |

|for a proportion? | |

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|What is the formula for calculating a| |

|confidence interval for a proportion?| |

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|Problem 8 – College Sporting Events |

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|A NCAA survey found that 2,486 of a sample of 10,904 college undergraduates said they attended a college sporting event over the last month. 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 attended a college |

|sporting event over the last month. |

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|How can the calculator help us find | |

|the confidence interval for | |

|proportions? | |

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|Problem 9 – Customer Service Sample Size |

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|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? |

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|Problem 10 – Customer Service Sample Size |

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|Suppose that you wanted to estimate p = the true proportion of shoppers at the mall who have a tattoo with 98% confidence and a margin of error of no more |

|than 0.10. How many mall shoppers should you survey? |

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|10.2 Estimating a Population Mean |

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|Since we don’t usually know the value| |

|of µ or σ, what do we use instead? | |

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|What type of distribution does this | |

|result in? | |

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|What are the characteristics of the | |

|t-distribution? | |

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|What three conditions need to be met |1. |

|to calculate a confidence interval | |

|for a population mean? | |

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|What is the formula for calculating a| |

|confidence interval for a population | |

|mean? | |

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|Problem 11 – the t distribution |

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|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? |

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

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|Problem 12 – NOX levels |

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|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. |

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|How can the calculator help us find | |

|the confidence interval for means? | |

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|Problem 13 – Homework claims |

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|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. |

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

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

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|What is a paired t procedure? | |

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|What are the different ways to | |

|conduct a paired t procedure? | |

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|Problem 14 – Caffeine dependence |

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|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. |

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|subject |

|deprscaf |

|deprsplac |

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|1 |

|5 |

|16 |

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|23 |

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|7 |

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|8 |

|14 |

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|6 |

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|24 |

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|0 |

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|0 |

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|2 |

|15 |

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|10 |

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|1 |

|0 |

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

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|What are the difference purposes of | |

|random selection and random | |

|assignment? | |

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|Problem 15 – Is the express lane faster? |

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|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 |

|express lane |

|(seconds) |

|Time in |

|regular lane |

|(seconds) |

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|337 |

|342 |

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|226 |

|472 |

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|502 |

|456 |

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|408 |

|529 |

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|151 |

|181 |

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|284 |

|339 |

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|150 |

|229 |

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|357 |

|263 |

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|349 |

|332 |

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|257 |

|352 |

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|321 |

|341 |

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|383 |

|397 |

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|565 |

|694 |

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|363 |

|324 |

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|85 |

|127 |

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|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 and Symbol for the Measure of Spread | | | |

|Condition: _____________ | | | |

|Condition: _____________ | | | |

|Condition: _____________ | | | |

|Formula for a confidence interval | | | |

|Generic Interpretation | |

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