3Topic 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 samplestatistic pˆ is called the standard error of pˆ.Confidence Interval for a population proportion :where n . P^ >= 10 and n (1-p^)>= 10Z * Critical value-Z is calculated based on level of confidenceWhen running for example 95% Confidence Interval:95% is called Confidence Level andwe are allowing possible 5% for error, we call this alpha (α )= 5% where α is the significant level
4Topic 16 - Confidence Interval: Proportion Click on STAT, TESTS and scroll down to1-PropZint…To calculate Confidence IntervalYou need to have x, n and C-Levelx and n comes from the samplePlease note if you have p-hat and n calculate x = p-hat * n, round your answer
6Watch OutA 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.
7The margin- of- error is affected by several factors primarilyA 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.
8Choosing the sample size The confidence interval for the a Normal population will have a specified margin of error m when the sample size isIf n is not a whole number then round up.
9Example: Activity 16-8: Cursive Writing The number of essays needed for a 99% CI is 0.01 = √[ (.15)(.85) /n]; n = (2.576 /.01)2 (.15)(.85) = ; n = Remember to round UPYou 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.
12Topic 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 hypothesisStep 3: Decide what test we have to run, in case of proportion, we use Z-test in proportion
13Topic 17 – Test of Significant: Proportion Step 4: Run the test from calculatorStep 5: From the calculator write down the p-value and Z-testStep 6: Compare your p-value with α – alpha – Significant LevelIf p-value is smaller than αwe “reject” the null hypothesis, then it is statistically significant based on data.If p-value is greater than the αwe “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 αp- value > 0.1: little or no evidence against H0• < p- value <= 0.10: some evidence against H0• < p- value <= 0.05: moderate evidence against H0• < p- value <= 0.01: strong evidence against H0• p- value <= 0.001: very strong evidence against H0
14Topic 17 – Test of Significant: Proportion Click on STAT, TESTS and scroll down to1-PropZTest…To calculate One Sample Proportion Z-TestYou need to have P0 , x, n and Alternative HypothesisP0 is π0 from Null Hypothesisx and n comes from the samplePlease note if you have p-hat and n calculate x = p-hat * n, round your answerProp is the alternative hypothesis
17Watch OutAlpha = 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.