Confidence Intervals and Hypothesis Testing Using and.

Two Statistical Procedures Confidence Intervals “which is an interval of values that the researcher is fairly “confident” covers the true value of the population.” Hypothesis Testing (or significance testing) “uses sample data to reject the notion that chance alone can explain the sample results.”

Remember Empirical Rule Going out 1·2.5, 2·2.5, 3·2.5 standard deviations from the mean, 69, gives us information regarding the amount of data contained in the range (interval).

Confidence Level P(-z* < Z < z*) =.90  z* = P(-z* < Z < z*) =.95  z* = P(-z* < Z < z*) =.99  z* = Confidence Level can vary. Since we want to be fairly sure about our confidence intervals, we usually use confidence levels in the 90+% range.

What is a Confidence Interval? It is a range of values calculated from a sample(s), based on the sample mean or proportion, sample standard deviation, sample size, and level of confidence. We use it to infer possible (plausible) values for the true population parameter (  or  ). Example: 95% CI for  is to Example: 90% CI for  is to 0.265

How is a Confidence Interval constructed? __%Confidence Interval = statistic  (critical value)(standard error of statistic) Example for Sample Proportion: Example for Sample Mean:

Margin of Error

95% CI of # of Tails/10 flips Sample Prop SE Prop 95.0 % C.I. T% ( , ) Interpretation: Based on this sample, we are 95% confident that the true population proportion (  ) of number of tails in 10 flips is between and

95% CI of My Age N MEAN STDEV SE MEAN 95 % C.I. Age (34.726, ) Interpretation: Based on this sample, we are 95% confident that the true population mean (µ) of my age is between 34.7 and 37.5.

Revisit CLT The first graph represents X (number of G’s in story). The second graph represents (average number of G’s in 3 attempts). What 3 important facts must you remember about CLT?

Properties of CI 1) Sample statistic is the center of CI for it is our best (and sometimes our only) estimate of the unknown population parameter of interest. 2) Larger the level of confidence, the wider the confidence interval. 3) The larger the sample size, the narrower the confidence interval. 4) Sampling distribution must be normally distributed (CLT, n and n(1- )  10).

How Many G’s? N MEAN STDEV SEMEAN AvgG % C.I. = = ± 1.997(1.02) = (59.423, )

95% CI of Brown M&Ms % Suppose you sampled a bag of 50 M&M’s and found 23 brown candies. Construct a 95% confidence interval. CI 95% = (0.322,.598)

95% CI of Brown M&Ms % 20 bags of M&M’s were used to calculate a sample proportion of brown candies per bag. Each sample proportion was used to construct a 95% CI which was graphed. How many CI do NOT contain 30%?

Various CI’s w/90% Conf. Level

Example Exam Questions Which of the following BEST describes what 95% confidence means in a 95% CI for  of (7.8, 9.4)? A.There is a 95% probability that  is between 7.8 and 9.4. B.In repeated sampling,  will fall between 7.8 and 9.4 about 95% of the time. C.In repeated sampling, 95% of the confidence intervals fall between 7.8 and 9.4. D.In repeated sampling, the confidence intervals will contain  about 95% of the time.

Hypothesis Testing Use hypothesis testing when you can state a hypothesis based on an assumption about the population parameter. Null hypothesis, H o, states that the population parameter is equal to an assumed true value. Alternative hypothesis, H a, states that the population parameter is, or  to an assumed true value. H o :  = 35 H a :  < 35

Example of Hypothesis Testing Is there a statistical difference between my age and students’ sample mean of my age? H o :  = 35 H a :  > 35 TEST OF MU = VS MU G.T N MEAN STDEV SEMEAN T P VALUE Age

What exactly is a p-value? 1. Bias in sampling 2. Luck of the draw—misrepresentative sample 3. Significant difference Indication of significance!

What exactly is a p-value? Is the probability, assuming H o is true, of getting data like this or ‘worse’ (more like the alternative hypothesis). Is the probability of obtaining extreme statistics or more when H o is true. You compare the p-value to alpha, . If p-value < , reject H o. If p-value > , fail to reject H o.

What exactly is a p-value in context of problem? Is the probability, assuming H o (my age is 35) is true, of getting data like this or ‘worse’ (more like the alternative hypothesis—claiming I am older than 35). Is the probability of obtaining extreme statistics (sample mean of from sample size of 69 students) or more when H o (my age is 35) is true. If alpha is 0.05 and p-value (probability) is 0.056, since p-value > alpha we will fail to reject that H o (my age is 35) is true.

Example of Hypothesis Testing Determine if the coin flipped was fair or not? H o : p =.5 H a : p .5 1PropZTest(not equal) = 356/690 =.5159 z =.8375 p-value =.4023 n = 690

Power, , and  Let confidence level be represented by L. Then 1 – L =  (OR 1 -  = L).  is called a Type I error.  is known as the Type II error. Power is equal to 1 - . All 3 of these have probability values (likelihood's of occurrence.).

Hypothesis Testing like a court case Assume innocent Null hypothesis is true—when person is actually innocent Null hypothesis is false—when person is actually guilty Fail to reject null hypothesis— found innocent Confidence Level (1-  )— found innocent, when innocent Type II Error  - -found innocent, when guilty Reject null hypothesis— found guilty Type I Error  - -found guilty, when innocent Power (1-  )— found guilty, when guilty

What are the Type I and II Errors? Your boss asked you to test a new marketing scheme to potentially increase profits for the company. But the new scheme also increases the expense of marketing the product. What are the two potential ERRORs in the context of this problem?

P-Value Question Which of the following statements is correct? A.A large p-value indicates that the data is consistent with the alternative hypothesis. B.The smaller the p-value, the weaker the evidence against the null hypothesis. C.The p-value measures the probability of making a Type II error. D.The p-value measures the probability that the hypothesis is true. E.An extremely small p-value indicates that the actual data differs from the expected if the null hypothesis were true.

Power Question Power in this situation refers to A.the ability to detect a difference when in fact there is no difference. B.the ability to not detect a difference when in fact there is no difference. C.the ability to detect a difference when in fact there is a difference. D.the ability to not detect a difference when in fact there is a difference. E.the ability to make a correct decision regardless if there is a difference or not.

Sample Question The government requires that water be sampled and tested for safety every day. If the worker responsible for these testing uses a null hypothesis that means the water is safe, which of the following statements correctly describes a Type I and Type II error? A.A Type I error would mean that the citizens were not warned of contaminated water, but a Type II error would mean that the citizens would panic although the water is ok. B.A Type II error would mean that the citizens were not warned of contaminated water, but a Type I error would mean that the citizens would panic although the water was ok. C.A Type II error would mean that the citizens were not warned of contaminated water, but a Type I error would mean that the citizens would not panic although the water is ok. D.A Type I error would mean that the citizens were not warned of contaminated water, but a Type II error would mean that the citizens would panic although the water is not ok. E.Both errors would cause unnecessary panic since a mistake means something went wrong but everything would work out.