Topics 16 - 18Unit 4 – Inference from Data: Principles

TOPIC 16 CONFIDENCE INTERVALS: PROPORTION

Topic 16 - Confidence Interval: Proportion

The estimated standard deviation of the sample

statistic pˆ is called the standard error of pˆ.n

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The purpose of confidence intervals is to use the sample statistic to construct an interval of values that you can be reasonably confident contains the actual, though unknown, parameter.

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Confidence Interval for a population proportion :

where n . P^ >= 10 and n (1-p^)>= 10

Z * Critical value-Z is calculated based on level of confidence

When running for example 95% Confidence Interval: 95% is called Confidence Level and we are allowing possible 5% for error, we call this alpha (α )= 5% where α is the significant level

Topic 16 - Confidence Interval: Proportion

Click on STAT, TESTS and scroll down to 1-PropZint…

To calculate Confidence IntervalYou need to have x, n and C-Level

x and n comes from the sample

Please note if you have p-hat and n calculate x = p-hat * n, round your answer

Exercise: 16-12: Credit Card Usage - Page 347

Exercise: 16-13: Responding to Katrina – Page 347

Watch Out

• A confidence interval is just that— an interval— so it includes all values between its endpoints.

• Do not mistakenly think that only the endpoints matter or that only the margin- of- error matters.

• The midpoint and actual values within the interval matter.

The margin- of- error is affected by several factorsprimarily • A higher confidence level produces a greater margin- of- error

( a wider interval). • A larger sample size produces a smaller margin- of- error

( a narrower interval). • Common confidence levels are 90%, 95%, and 99%. • Always check the technical conditions before applying this

procedure. • The sample is considered large enough for this procedure to be

valid as long as npˆ>= 10 and n(1 –pˆ) >=10. If this condition is not met, then the normal approximation of the sampling distribution is not valid and the reported confidence level may not be accurate.

• Always consider how the sample was selected to determine the population to which the interval applies.

Choosing the sample size

The confidence interval for the a Normal population will have a specified margin of error m when the sample size is

If n is not a whole number then round up.

PPmzn ˆ1ˆ

2*

Example: Activity 16-8: Cursive Writing

a. The number of essays needed for a 99% CI is0.01 = 2.576 √[ (.15)(.85) /n]; n = (2.576 /.01)2 (.15)(.85) = 8460.614; n = 8461 Remember to round UP

b. You could use a lower confidence level (95% or 90% confidence, for example), or you could use a wider margin-of-error, say .02. Either of these choices would allow you to select a smaller (random) sample.

PPmzn ˆ1ˆ

2*

Activity 16-11: Penny Activities - Page 347

PPmzn ˆ1ˆ

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TOPIC 17 - TESTS OF SIGNIFICANCE: PROPORTIONS

Topic 17 – Test of Significant: ProportionA sample result that is very unlikely to occur by random chance alone is said to be statistically significant. We now formalize this process of determining whether or not a sample result provides statistically significant evidence against a conjecture about the population parameter. The resulting procedure is called a test of significance.

A significance test is designed to assess the strength of evidence against the null hypothesis.

Step 2: we initiate hypothesis regarding the question – we can not run test of significant without establishing the hypothesis

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Step 3: Decide what test we have to run, in case of proportion, we use Z-test in proportion

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Step 1: Identify and define the parameter.

Topic 17 – Test of Significant: Proportion

Step 4: Run the test from calculator

Step 5: From the calculator write down the p-value and Z-test

Step 6: Compare your p-value with α – alpha – Significant Level

If p-value is smaller than α we “reject” the null hypothesis, then it is statistically significant based on data.

If p-value is greater than the α we “Fail to reject” the null hypothesis, then it is not statistically significant based on data.

Last step: we write conclusion based on step 6 at significant level α

• p- value > 0.1: little or no evidence against H0 • 0.05 < p- value <= 0.10: some evidence against H0 • 0.01 < p- value <= 0.05: moderate evidence against H0 • 0.001 < p- value <= 0.01: strong evidence against H0 • p- value <= 0.001: very strong evidence against H0

Topic 17 – Test of Significant: Proportion

Click on STAT, TESTS and scroll down to 1-PropZTest…

To calculate One Sample Proportion Z-TestYou need to have P0 , x, n and Alternative Hypothesis

P0 is π0 from Null Hypothesis

x and n comes from the samplePlease note if you have p-hat and n calculate x = p-hat * n, round your answer

Prop is the alternative hypothesis

Exercise 17-6: Properties of p-value – Page 371Exercise 17-7: Properties of p-value – Page 371 Exercise 17-8: Wonderful Conclusions– Page 371

Exercise 17-12: Kissing Couples – Page 372Exercise: 17-26: Employee Sick Days–Page 375 Exercise: 17-27: Stating Hypothesis –Page 375

TOPIC 18 MORE INFERENCE CONSIDERATION

Watch Out

• Alpha = αA Type I error is sometimes referred to as a false alarm because the researcher mistakenly thinks that the parameter value differs from what was hypothesized.

• Beta = βa Type II error can be called a missed opportunity because the parameter really did differ from what was hypothesized, yet the researchers failed to realize it.

• 1 – βThe power of a statistical test is the probability that the null hypothesis will be rejected when it is actually false ( and therefore should be rejected). Particularly with small sample sizes, a test may have low power, so it is important to recognize that failing to reject the null hypothesis does not mean accepting it as being true.