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Chapter 8

Chapter 8. Estimating with Confidence. 8.1 Confidence Intervals: The Basics. - point estimator : is a statistic that provides an estimate of a population parameter - point estimate : value of that statistic from a sample To solve in calculator for x̅:

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Chapter 8

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  1. Chapter 8 Estimating with Confidence

  2. 8.1 Confidence Intervals: The Basics -point estimator: is a statistic that provides an estimate of a population parameter -point estimate: value of that statistic from a sample To solve in calculator for x̅: Put in data in L1 (stat, edit) CALC, 1-VAR STATS X̅:30.35 σ²:13.879=192.63 The Idea of Confidence Interval -C% confidence interval: gives an interval plausible values for a parameter point estimate ∓ margin of error

  3. 8.1 CONT. -margin of error: difference between point estimate and the true parameter value will be less than this -confidence level C: overall success rate of the method of calculating the confidence interval Interpreting Confidence Intervals and Confidence Levels “We are C% confident that the interval from _____ to _____ captures the (context of parameter)” • We are 95% confident that the interval from 0.63 to 0.69 captures the true proportion of US adults that favor an amendment to the Constitution. • p̂=0.63+0.63 / 2 =0.66 m= 0.69-0.63 / 2 = 0.03

  4. 8.2 Estimating a Population Proportion Conditions • Random-The Data should come from a well-designed random sample or randomized experiment • 10%- n ≤ 1/10N • Large counts- both np̂ and n(1-p̂) are at least 10 Identify conditions met: Random: SRS Large Count: 1100( 775/1100) ≥ 10 1100(1-775/1100)≥ 10 10%: 1100 is less than 10% of population

  5. 8.2 cont Confidence Interval STAT TEST 1-PROPZINT: X:246 N:439 C-LEVEL: 0.95 STAT TEST 1-PROPZINT: X:427 C:0.99 (0.13835,0.17413) N:2733 p̂: 0.15623 N:2733 (0.51393,0.60679) P̂: 0.5603644 N:439

  6. 8.3 Estimating a Population Mean Standard error : N-1 degree of freedom: is used to specify a particular t distribution. Degrees of freedom are found by subtracting 1 from the sample size n T distribution: calculator part Large sample: The population distribution is Normal or the sample size is large (n>30)

  7. Open Ended Questions A consumer watchdog organization estimates the mean weight of 1 ounce of “Fun Size” candy bars to see if customers are getting full value for their money. A random sample of 25 bars is selected and weighed, and the organization reports that a 90% confidence interval for the true mean weight of the candy bars is 0.992 to 0.998 ounces. • Interpret the 90% confidence interval 0.992 to 0.998 in the context of the problem? • What is the point estimate and the margin of error in the context of the problem?

  8. Open Ended Questions A simple random sample of 1100 males aged 12 to 17 in the United States were asked whether they played massive multiplayer online role-playing games (MMORPGs); 775 said that they did. We want to use this information to construct a 95% confidence interval to estimate the proportion of all U.S. males aged 12 to 17 who play MMORPGs. (a) State the parameter our confidence interval will estimate. (b) Identify each of the conditions that must be met to use this procedure, and explain how you know that each one has been satisfied. (c) Find the appropriate critical value and the standard error of the sample proportion.

  9. Open Ended Question An insect ecologist reports a 95% confidence interval for the mean length of full-grown aquatic larvae of the Phantom Midge Chaoborusalbatus to be 6.9 to 8.5 mm, based on a sample of 9 individual larvae. a)What are the point estimate and margin of error associated with this confidence interval? b)The ecologist stated that “all necessary conditions for constructing this confidence interval were met.” What are the conditions? c)If the ecologist had reporated a 99% confidence interval instead of a 95%, how would it have been different? Explain.

  10. MC Questions An airline observes a random sample of its flights on a particular route. The 95% confidence interval for the mean time (in minutes) for all flights on this route is calculated to be (47.0, 53.0). Which of the following is NOT true? • At the 95% confidence level, the true mean flight time is within 3.0 minutes of the sample mean flight time. • If the true mean flight time were outside the interval (47.0, 53.0) then the sample mean that was found would be very unlikely. • Approximately 2 ½% of flights on this route are longer than 53 minute. • We are 95% confident that the true mean flight time is between 47.0 and 53.0 minutes • If many random sample of the same size were taken and the 95% confidence interval were calculated, then 95% of the confidence intervals would contain the true mean flight time

  11. MC Questions A marketing consultant wants to estimate the proportion of all shoppers at a certain mall who make at least one purchase. He stands at a mall exit for two hours on a weekday afternoon and flips a coin each time a shopper leaves. If the coin comes up heads, he asks them if they have made any purchases during this visit. After two hours, he has 132 responses, 104 of whom made a purchase. Which condition for constructing a confidence interval for a proportion has the consultant failed to satisfy? (a) npˆ ≥10 (b) n p (1 10 − ≥ ˆ) (c) n ≥ 30 (d) The data is a random sample from the population of interest. (e) The sample is less than 10% of the population.

  12. MC Questions Find the critical t* value for 95% confidence interval with 8 degrees of freedom (a)2.603 (b)1.006 (c) 2.306 (d)3.980

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