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BIOL499A (11)
Chapter 19-21

Chapter 19-21 copy.pdf

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Department
Biology (Biological Sciences)
Course Code
BIOL499A
Professor
Blaine Mullins

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Chapters 19 - 21: Statistical Inference – Confidence Interval and Hypothesis Testing for One Population Proportion In these chapters, we will talk about:  Estimation of a population proportion,  Large-sample confidence interval for a population proportion,  How to choose the sample size for a study about a population proportion,  One-sample hypothesis test for a population proportion when sample size is large. Goal:  One population proportion, p  Two population proportions, p  p 1 2  One population mean,   Two population means,   1 2  More than two population proportions  More than two population means Notations:  Suppose Z has a standard normal distribution. Then, for given * *  (0  1) ,  z(or for simplisity is defined as: * PZ()  z   Definition: a100(1)% confidence interval (C.I.) for a parameeis an interval (L, U) such that 1 PL ( )    L and U are called lower and upper bound of the confidence interval, respectively. The percentage 100(1)% is called confidence level. The presence of bacteria in a urine sample (bacteriuria) is sometimes associated with symptoms of kidney disease. It was found in a study 30 out of 500 were positive for bacteriuria. Find a 95% confidence interval for the proportion of people who are positive for bacteriuria. Confidence Interval for One Population Proportion: A 100(1)%approximate confidence interval for populatiop, is givenn, by 1 where , pz  ˆ*p(1-ppz ˆ ˆ pqq p n n pp1 ) ME  z * The quantity n is called the margin of error. Example 1: An analyst wishes to estimate the market share captured by Brand X detergent - that is, the proportion of Brand X sales compared to the total sales of all detergents. From data supplied by several stores, the analyst finds that out of a total of 325 boxes of detergent sold, 120 were BranX. (a) Estimate the market share captured by BrandX. (b) Estimate the standard error of the above estimate. (c) Find a 95% confidence interval for . p Example 2: A sample of 78 university students revealed that 49 carried their books and notes in a backpack. (a) Estimate the population proportion of students who carry their books and notes in a backpack. (b) Obtain the estimated standard error of the above estimate. (a) Give an approximate 95.4% error margin. (b) Find an approximate 95.4% confid ence interval for the population proportion of students who carry their books and notes in a backpack. Top-Hat Question (Review 9-1): Example 3: The table below summarizes the result of a survey of opinions of both smokers and nonsmokers on an ordinance prohibiting smoking in bars. AgSaippfrrent Smokers 16 20 111 Nonsmokers 90 349 114 Construct a 99% confidence interval for the proportion of smokers who are against the ban in bars. What is upper limit of the confidence interval? Sample Size for a Study about a Population Proportion: To determine how large a sample is needed for estimating alation proportion, we must specify: ME = the desired margin of error 1 = the probability associated with the error margin. Then, the formula for calculating the sample nize, , is given by * 2 n pz (1 ) ME Note that: pis unknown. Hence * 2 z pn p  ˆˆ (1 ) E  If the value of p is known to be roughly in the neighborhood of a * value p , thenncan be determined from * 2 pp z * (1 ) ME  We may take a small-scale preliminary sampling is to obtain an estimate of p to be used in the formula to computen. Therefore, * 2 pn p z ˆ(1 ) E  Without prior knowledge of p, the value of p(1 p) can be replaced by its maximum possiblevalue 0.5(1-0.5) and can bn determined from * 2 n  z 0.5 (1 0.5) E Example 4: A study is conducted to evaluate the proportion of infected needles used by injection drug users. (a) If there is no preliminary study is available, how large a random sample is needed to form a 95 percent conf idence interval with an error bound of 5 percent? (b) A preliminary study of a random sample of 160 needles that were used before a needle exchange program was established found 108 to be HIV-positive. In the context of this study, how large a sample is needed to form a 95 percent confidence interval on the population proportion of pre-exchange-program needles that are HIV- positive, with an error bound of 5 percent? Example 5: A political pollster wants to estimate the percentage of voters who will vote for the Democratic candidate in a presidential campaign. The pollster wishes to have 90% confidence that her prediction is correct to within  4% of the population proportion. (a) What sample size is needed? Top-Hat Question (Review 9-2): (b) If the pollster wants to have 95% confidence, what sample size is needed? (c) If she wants to have a 95% confid ence and a sampling error of 3%, what sample size is needed? The presence of bacteria in a urin e sample (bacteriuria) is sometimes associated with symptoms of kidney disease.  Assume that a determination of bacteriuria has been made over a large population at one point in time and that 4% of the population are positive for bacteriuria.  However, the general feeling is that the rate of bacteriuria has increased.  To test this feeling, a new study by mail questionnaire was performed and it was found that 30 out of 500 were positive for bacteriuria.  Can we conclude that the true rate of bacteriuria has increased? Intereste
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