Oct 16 2007 LEC 6 ECON 140A 240A 1 Interval Estimation and Hypothesis Testing L Phillips I Introduction From a simple random sample of voters we obtain a sample proportion of the voters supporting a candidate but we do not know the proportion for the entire population of voters Can we say that this population proportion lies in a specified interval with some likelihood Of course the candidate hopes this interval is above 0 5 and that the likelihood is nearly certain Recall that our series of monthly rates of return for the UC stock index consisted of values that were independent of one another From this data we can calculate a sample mean but we do not know the mean rate of return for the underlying population generating these monthly observations Once again can we say the population mean lies in some interval and on average if we used this procedure many times what fraction of the time we would be right II Confidence Intervals for Sample Proportions and Population Proportions Suppose a simple random sample of 1000 California Democratic likely voters produces a sample proportion of 0 53 who support Hillary Does the interval for the population proportion lie above 0 50 What fraction of the time if we used this procedure again and again would this interval be correct We know that proportions are distributed binomially and since this is a large sample we can approximate it with the normal distribution We also know that for the normal distribution approximately 68 percent of the observations lie within one standard deviation of the mean and that 95 of the observations lie within plus or minus 1 96 standard deviations of the mean i e P 1 96 p p p 1 96 0 95 1 Oct 16 2007 LEC 6 ECON 140A 240A 2 Interval Estimation and Hypothesis Testing where p p 1 p L Phillips n With some manipulation multiply the inequality by p and add p to the inequality we can restate this as P p 1 96 p p p 1 96 p 0 95 2 This means that this sampling procedure if repeatedly used would produce sample proportions that would lie within plus or minus 1 96 standard deviations from the population mean 95 of the time Alternatively Eq 1 can be expressed as P p 1 96 p p p 1 96 p 0 95 3 which can be obtained by multiplying the inequality in Eq 1 by p and subtracting p and multiplying by minus one which reverses the inequality This expression in Eq 3 provides an interval within which the true population proportion will lie 95 of the time However since we do not know the population proportion p we use the sample proportion p to calculate the standard deviation s p 1 p n p 4 for use in Eq 3 P p 1 96s p p p 1 96 s p 0 95 5 As a numerical example take the sample mean of 0 53 for a sample of 1000 The standard deviation is s p 0 53 1 0 53 1000 0 0158 6 The 95 confidence interval for the population proportion of California voters supporting Hillary is Oct 16 2007 LEC 6 ECON 140A 240A 3 Interval Estimation and Hypothesis Testing P 0 499 p 0 561 0 95 L Phillips 7 A politician supporting Hillary might be pleased with the interval which mostly lies above 0 50 but nervous that 5 of the time the population proportion could lie outside of this interval especially the proportion that lies below 0 5 What sort of interval for p would have a 99 confidence level P p 2 58 s p p p 2 58 s p 0 99 8 Which calculates to P 0 489 p 0 571 0 99 9 This wider confidence interval extends even more above 0 50 but also more below it making a politician supporting Hillary nervous The only way we could do better is with a larger sample size i e with a higher cost to obtain the larger sample of voters Even in that case there is no guarantee the sample proportion will be 0 53 or higher It could be lower given the random nature of the sample Conditional on a value for the sample proportion these formulae could be used to see how the standard deviation and the confidence interval will vary with sample size given the level of confidence chosen III Confidence Intervals for Sample Means and Population Means Using our example of a sample of twelve monthly rates of return for the UC stock index fund with sample mean 1 61 and standard deviation 4 04 the variable t t 1 61 4 04 12 has the t distribution for eleven degrees of freedom One degree of freedom in this sample of twelve observations has been used in the calculation of the sample mean which in turn is used to calculate the sample standard deviation Oct 16 2007 LEC 6 ECON 140A 240A 4 Interval Estimation and Hypothesis Testing r j r 2 n 1 1 2 s j L Phillips 10 For eleven degrees of freedom with a probability of 0 95 the population mean falls in the interval P r t0 025 4 04 12 r t0 025 4 04 12 0 95 11 For eleven degrees of freedom t0 025 is 2 20 i e 2 5 percent of the distribution lies above t 2 2 and 2 5 percent of the distribution lies below t 2 2 For our sample mean r 1 61 the 95 confidence interval for the population mean is P 0 96 4 18 0 95 12 This interval for the mean monthly rate of return for the population is quite broad and a larger sample would likely help Compare this value of t of 2 2 for eleven degrees of freedom to obtain a 95 confidence interval with the value of z 1 96 from the normal distribution to obtain a 95 confidence interval Ignorance has its price and not knowing the variance of the population of monthly returns forces us to calculate the sample deviation and to use the t distribution IV Hypothesis Tests for Proportions There are four steps to statistically testing a hypothesis The first step is to formulate all of the hypotheses both the null or maintained hypothesis and the alternative hypothesis The second step is to identify a test statistic that will assess the evidence against the null hypothesis The third step is a probability statement that answers the question if the null hypothesis were true then what is the probability of observing a test statistic at least as extreme as the one observed The fourth step is to compare this probability to some chosen critical level of significance say 5 This Oct 16 2007 LEC 6 ECON 140A 240A 5 Interval Estimation and Hypothesis Testing L Phillips critical level i e a willingness to bear the burden of a probability of rejecting the null hypothesis even if it were true as high as 5 is designated An example of a hypothesis …
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