Intro to Confidence Intervals SECTION 10.1 1 Confidence Intervals - - PDF document

Confidence Intervals Slides.notebook 1 December 22, 2015

Intro to Confidence Intervals

SECTION 10.1

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Definitions

• Statistical Inference:
• estimate of the characteristics of a population
• derived from the analysis of a sample drawn

from the population

• provides methods of drawing conclusions

about a population from sample data

• population: everyone
• parameter: a number from a population
• Symbols­­­
• sample: a part of the population
• statistic: a number from a sample
• Symbols­­­

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NOTE TO SELF

The Central Limit theorem plays an important role in statistical inference. 2 types of statistical inference: Confidence intervals (10.1) Tests of significance (10.2)

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• You want to estimate the mean SAT Math score for the more than 350,000 high

school seniors in California.

• You give the test to a simple random sample (SRS) of 500 CA seniors.
• RECALL:
• The central limit theorem tells us that the mean, x, of 500 scores has a

distribution that is close to normal.

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• The mean of this normal sampling distribution is the

same as the unknown mean, , of the entire population.

• The standard deviation of x for an SRS of 500

students is , where is the standard deviation of individual SAT MATH scores among all CA high school seniors.

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Statistical Confidence

• The 68­95­99.7 rule says that in 95% of all samples, the

mean score x for the sample will be within 2 standard deviations (9 points in this example) of the population mean score

• Therefore, in 95% of all samples, the unknown lies

between x – 9 and x + 9.

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Confidence Interval has the form:

estimate + margin of error Estimate: guess for unknown parameter (usually μ) We will use X here MOE: how accurate we believe guess is based on variability of estimate

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Confidence Level, C

Gives probability that the interval will capture true population mean, μ, in repeated samples “C” will be expressed in decimal form Note: we generally want a confidence level of .9 or 90%

• r higher

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• This shows the result of drawing 25 SRSs from the same population

and calculating a 95% confidence interval from each sample.

• The center x of each interval is marked by a dot.
• The arrows on either side of the dot span the confidence interval.
• CONCLUSION: 95% of all samples give an interval that contains the

population mean .

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Formula for Confidence Interval

Choose an SRS of size n from a population having unknown mean μ and known standard deviation σ. A level C confidence interval for μ is

estimate + margin of error

• Everything will be given in the problem except z*
• z* depends on the confidence level you choose
• z* is the value with area C between –z* and z*

under the standard normal curve.

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Most common confidence levels: Calculator can find z* values:

C Tail Area z* 90% 0.05 1.645 95% 0.025 1.960 99% 0.005 2.576

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What does 95% confidence mean?

• We are 95% confident that the true

population mean is captured in the interval OR

• We are using a procedure that captures the

true population mean 95% of the time.

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Example 1

I sample 30 cats. X = 6 lbs. I know that σ = 3 lbs.

• a) Find a 90% CI.
• b) Find a 95% CI.
• c) Find a 99% CI.

**Express what these confidence intervals indicate in terms of the problem.** e) Explain why the intervals get wider as your confidence level increases.

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CALCULATOR!!!!

• STAT
• TEST
• Z­INTERVAL (#7)
• STATS
• Plug info in and calculate.
• Show

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Example 2

Heights are normally distributed. I measure the heights

• f 25 randomly selected students. The average is 66
• inches. I know that the population S.D. (σ) is 4 inches.
• a) Find a 90% CI.
• b) Find a 95% CI.
• c) Find a 97% CI.
• d) Find a 99.9% CI

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Example 3

SAT verbal scores are normally distributed with σ = 50. 25 students are selected randomly and their average score is 492.

• a) Find a 99% Confidence Interval.
• If 100 students are selected and their average is still

492, find the new 99% Confidence Interval.

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Example 4

The following 10 bunny weights are collected. 7 8 9 8.5 10 11 8.5 9.5 10.5 10 Find a 90% confidence interval for bunny weights based on this sample! Assume σ = 1.5.

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In Summary

Increasing my confidence level will _________________ my margin of error, and therefore also ____________ my interval width. Increasing my sample size will _______________ my margin of error. If we want to cut our M.O.E. in half, what should we do to our sample size?

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