# BPS - 5th Ed. Chapter 191 Inference about a Population Proportion.

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BPS - 5th Ed. Chapter 191 Inference about a Population Proportion

BPS - 5th Ed. Chapter 192 u The proportion of a population that has some outcome (success) is p. u The proportion of successes in a sample is measured by the sample proportion: Proportions p-hat

BPS - 5th Ed. Chapter 193 Inference about a Proportion Simple Conditions

BPS - 5th Ed. Chapter 194 Inference about a Proportion Sampling Distribution

BPS - 5th Ed. Chapter 195 Case Study Science News, Jan. 27, 1995, p. 451. Comparing Fingerprint Patterns

BPS - 5th Ed. Chapter 196 Case Study: Fingerprints u Fingerprints are a sexually dimorphic trait…which means they are among traits that may be influenced by prenatal hormones. u It is known… –Most people have more ridges in the fingerprints of the right hand. (People with more ridges in the left hand have leftward asymmetry.) –Women are more likely than men to have leftward asymmetry. u Compare fingerprint patterns of heterosexual and homosexual men.

BPS - 5th Ed. Chapter 197 Case Study: Fingerprints Study Results u 66 homosexual men were studied. 20 (30%) of the homosexual men showed leftward asymmetry. u 186 heterosexual men were also studied. 26 (14%) of the heterosexual men showed leftward asymmetry.

BPS - 5th Ed. Chapter 198 Case Study: Fingerprints A Question Assume that the proportion of all men who have leftward asymmetry is 15%. Is it unusual to observe a sample of 66 men with a sample proportion ( ) of 30% if the true population proportion (p) is 15%?

BPS - 5th Ed. Chapter 199 Case Study: Fingerprints Sampling Distribution

BPS - 5th Ed. Chapter 1910 Case Study: Fingerprints Answer to Question u Where should about 95% of the sample proportions lie? –mean plus or minus two standard deviations 0.15 2(0.044) = 0.062 0.15 + 2(0.044) = 0.238 –95% should fall between 0.062 & 0.238 u It would be unusual to see 30% with leftward asymmetry (30% is not between 6.2% & 23.8%).

BPS - 5th Ed. Chapter 1911 u Inference about a population proportion p is based on the z statistic that results from standardizing : Standardized Sample Proportion –z has approximately the standard normal distribution as long as the sample is not too small and the sample is not a large part of the entire population.

BPS - 5th Ed. Chapter 1912 Building a Confidence Interval Population Proportion

BPS - 5th Ed. Chapter 1913 Since the population proportion p is unknown, the standard deviation of the sample proportion will need to be estimated by substituting for p. Standard Error

BPS - 5th Ed. Chapter 1914 Confidence Interval

BPS - 5th Ed. Chapter 1915 Case Study: Soft Drinks A certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis. A random sample of 100 customers yielded 18 who did in fact drink another brand of soft drink on a regular basis. Compute a 95% confidence interval (z* = 1.960) to estimate the proportion of interest.

BPS - 5th Ed. Chapter 1916 Case Study: Soft Drinks We are 95% confident that between 10.5% and 25.5% of the soft drink bottlers customers drink another brand of soft drink on a regular basis.

BPS - 5th Ed. Chapter 1917 u The standard confidence interval approach yields unstable or erratic inferences. u By adding four imaginary observations (two successes & two failures), the inferences can be stabilized. u This leads to more accurate inference of a population proportion. Adjustment to Confidence Interval More Accurate Confidence Intervals for a Proportion

BPS - 5th Ed. Chapter 1918 Adjustment to Confidence Interval More Accurate Confidence Intervals for a Proportion

BPS - 5th Ed. Chapter 1919 Case Study: Soft Drinks Plus Four Confidence Interval We are 95% confident that between 12.0% and 27.2% of the soft drink bottlers customers drink another brand of soft drink on a regular basis. (This is more accurate.)

BPS - 5th Ed. Chapter 1920 Choosing the Sample Size Use this procedure even if you plan to use the plus four method.

BPS - 5th Ed. Chapter 1921 Case Study: Soft Drinks Suppose a certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis using a 99% confidence interval, and we are instructed to do so such that the margin of error does not exceed 1 percent (0.01). What sample size will be required to enable us to create such an interval?

BPS - 5th Ed. Chapter 1922 Case Study: Soft Drinks Thus, we will need to sample at least 16589.44 of the soft drink bottlers customers. Note that since we cannot sample a fraction of an individual and using 16589 customers will yield a margin of error slightly more than 1% (0.01), our sample size should be n = 16590 customers. Since no preliminary results exist, use p* = 0.5.

BPS - 5th Ed. Chapter 1923 The Hypotheses for Proportions u Null: H 0 : p = p 0 u One sided alternatives H a : p > p 0 H a : p < p 0 u Two sided alternative H a : p p 0

BPS - 5th Ed. Chapter 1924 Test Statistic for Proportions u Start with the z statistic that results from standardizing : u Assuming that the null hypothesis is true (H 0 : p = p 0 ), we use p 0 in the place of p:

BPS - 5th Ed. Chapter 1925 P-value for Testing Proportions u H a : p > p 0 v P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. u H a : p < p 0 v P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. u H a : p p 0 v P-value is two times the probability of getting a value as large or larger than the absolute value of the observed test statistic (z) value.

BPS - 5th Ed. Chapter 1926

BPS - 5th Ed. Chapter 1927 Case Study Brown, C. S., (1994) To spank or not to spank. USA Weekend, April 22-24, pp. 4-7. Parental Discipline What are parents attitudes and practices on discipline?

BPS - 5th Ed. Chapter 1928 u Nationwide random telephone survey of 1,250 adults that covered many topics u 474 respondents had children under 18 living at home –results on parental discipline are based on the smaller sample u reported margin of error –5% for this smaller sample Case Study: Discipline Scenario

BPS - 5th Ed. Chapter 1929 The 1994 survey marks the first time a majority of parents reported not having physically disciplined their children in the previous year. Figures over the past six years show a steady decline in physical punishment, from a peak of 64 percent in 1988. –The 1994 sample proportion who did not spank or hit was 51% ! –Is this evidence that a majority of the population did not spank or hit? (Perform a test of significance.) Case Study: Discipline Reported Results

BPS - 5th Ed. Chapter 1930 Case Study: Discipline u Null: The proportion of parents who physically disciplined their children in 1993 is the same as the proportion [p] of parents who did not physically discipline their children. [H 0 : p = 0.50] u Alt: A majority (more than 50%) of parents did not physically discipline their children in 1993. [H a : p > 0.50] The Hypotheses

BPS - 5th Ed. Chapter 1931 Case Study: Discipline Based on the sample u n = 474 (large, so proportions follow Normal distribution) u no physical discipline: 51% – – standard error of p-hat: (where.50 is p 0 from the null hypothesis) u standardized score (test statistic) z = (0.51 - 0.50) / 0.023 = 0.43 Test Statistic

BPS - 5th Ed. Chapter 1932 Case Study: Discipline P-value = 0.3336 From Table A, z = 0.43 is the 66.64 th percentile. z = 0.43 0.5000.5230.4770.5460.4540.5690.431 012-23-3 z: P-value

BPS - 5th Ed. Chapter 1933 1. Hypotheses:H 0 : p = 0.50 H a : p > 0.50 2. Test Statistic: 3. P-value:P-value = P(Z > 0.43) = 1 – 0.6664 = 0.3336 4. Conclusion: Since the P-value is larger than = 0.10, there is no strong evidence that a majority of parents did not physically discipline their children during 1993. Case Study: Discipline