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. ^ p1 p n ^ The estimated standard deviation of the sample statistic pˆ is called the standard error of pˆ. Confidence Interval for a population proportion : ^ p 1 p n ^ ^ est imat e margin of error p Z * 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 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 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 2 z ˆ n P 1 Pˆ m * If n is not a whole number then round up. Example: Activity 16-8: Cursive Writing 2 z ˆ n P 1 Pˆ m * a. The number of essays needed for a 99% CI is 0.01 = 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. Activity 16-11: Penny Activities - Page 347 2 z ˆ n P 1 Pˆ m * TOPIC 17 - TESTS OF SIGNIFICANCE: PROPORTIONS 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 H 0 : H : a 0 0 0 or or 0 Step 3: Decide what test we have to run, in case of proportion, we use Z-test in proportion p Z 0 0 (1 0 ) n 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 371 Exercise 17-7: Properties of p-value – Page 371 Exercise 17-8: Wonderful Conclusions– Page 371 Exercise 17-12: Kissing Couples – Page 372 Exercise: 17-26: Employee Sick Days–Page 375 Exercise: 17-27: Stating Hypothesis –Page 375 TOPIC 18 MORE INFERENCE CONSIDERATION Watch Out • Alpha = α A 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 = β a 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–β The 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.