Inferential Statistics

Coin Flip

How many heads in a row would it take to convince you the coin is unfair? 1? 10?

Number of TossesApprox Probability of All Heads 1(½) 1 =.5 2(½) 2 =.25 3(½) 3 =.125 4(½) 4 =.063 5(½) 5 =.031 6(½) 6 =.016 7(½) 7 =.008 8(½) 8 =.004 9(½) 9 = (½) 10 = (½) 100 =7.88-e31

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Inferential Statistics To draw inference from a sample about the properties of a population Population distribution: The distribution of a given variable(parameter) for the entire population Sample distribution: A sample of size n, is drawn from the population and the variable’s distribution is called the sample distribution. Sampling distribution: This refers to the properties of a particular test statistic. The sampling distribution draws the distribution of the test statistic if it were calculated from a sample of size n, then resample using n observation to calculate another test statistic. Collect these into the sampling distribution.

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Law of Large Numbers and Central Limit Theorem How can we use this information? We can use our knowledge of the sampling distribution of a test statistic, a single realization of that test statistic to infer the probability that it came from a certain population

One Sample T-test of mean If the calculate Z statistic is large than the critical value (C.L.) then we reject the null hypothesis, we can also use p-values. That is the exactly probability of drawing a this sample from a population as is hypothesized under the null distribution. If the p-value is large (generally larger than.05 (5%)), we fail to reject the null, if it is small we reject the null. Z distribution (standard normal) vs. t-distribution (students t) The t distribution is used in situations where the population variance is unknown and the sample size is less than 30.

Hypothesis testing Develop a hypothesis about the population, then ask does the data in our sample support the hypothesized population characteristic. Ho: Null hypothesis Ha: Alternative hypothesis Significance level. The a critical point where the probability of realizing this sample when pulled from a population as hypothesized under the under the

Type I and II Errors (Innocent until proven Guilty) What if Ho = innocent alpha = the nominal size of the test (probability of a type I error) Beta = probability of a type II error 1-beta= the power of a test (ability to reject a false null)

Confidence Intervals Confidence intervals for the mean/proportion The population mean lies within the range.

Z(T-Test) of proportion where

Example: – Males represent 47.9% of the population over the age of 18.

Categorical/Categorical Crosstabulations (2 way frequency tables, Crosstabs, Bivariate distributions)

Chi-squared test of independence categorical/categorical with degrees of freedom (R-1)(C-1) where R = number of rows and C= number of columns

χ 2 =1.01 and the critical value with 1 degree of freedom at the 5% level is 3.84 fail to reject H0: The variables are independent, that is to say knowledge of one will not help to predict the outcome of the other

Categorical/Continuous Any statistic that applied to cont. variables done for each category – Mean, median, mode. – Variance, Std dev, skewness, kurtosis

Comparison of Means Z test (T-test) comparison of means. Null hypothesis is that the mean difference is 0

Where is the pooled estimate of the standard error of the mean, assuming the underlying population variances are equal. Pooled estimate of the standard error (population variances equal)

Continuous/Continuous Simple Correlation coefficient (Pearson’s product-moment correlation coefficient, Covariance) this ranges from +1 to -1

T-Test of correlation coefficient

Four sets of data with the same correlation of 0.816