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### Topics 16 - 18

Unit 4 – Inference from Data: Principles

Topic 16 Confidence Intervals: Proportion

Topic 16 - Confidence Interval: Proportion

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.

The estimated standard deviation of the sample

statistic pˆ is called the standard error of pˆ.

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 Interval

You 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 347Exercise: 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 factors

primarily

- 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.

Example: Activity 16-8: Cursive Writing

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

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.

Activity 16-11: Penny Activities - Page 347

Topic 17 – Test of Significant: Proportion

- A 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 1: Identify and define the parameter.

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

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

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-Test

You need to have P0 , x, n and Alternative Hypothesis

P0 is π0 from Null Hypothesis

x and n comes from the sample

Please 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 371Exercise 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.

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