Section 7.2 Confidence Interval For A Proportion
Section 7.2Confidence Interval for a ProportionBefore any inferences can be made about a proportion, certain conditions must be satisfied:1. The sample must be an SRS from the population of interest.2. The population must be at least 10 times the size of the sample.3. The number of successes must be npˆ 10 , and the number of failures must ben(1 pˆ ) 10 .The sample statistic for a population proportion is p̂ , so based on the formula for a CI, we havepˆ margin of error .---------------Width of the Confidence Interval-------------------margin of error margin of errorMeanorProportionSo if you have a confidence interval with width 0.22, then what is the margin of error?How do we find the margin of error if it is not given to us?The margin of error is computed using the critical value (a number based on our level ofconfidence) and the standard deviation (standard error) of the statistic.Critical Value: When the distribution is assumed to be normal, our critical value is found usingqnorm in R. If it is not normal, we will use the t distribution (discussed later). 1 confidence level The formula is: z * qnorm 2 Standard Deviation/Error: When working with proportions, the standard deviation of thep (1 p ). Since p is unknown, we will use the standard error. To calculate thestatistic p̂ isnpˆ (1 pˆ ).standard error of p̂ , use the formulanSo, pˆ margin of error pˆ z * pˆ (1 pˆ )nSection 7.2 – Confidence Interval for a Proportion1
Facts about Confidence Intervals For smaller n, the confidence interval becomes wider.For larger n, the confidence interval becomes narrower.Increasing the variance or increasing the confidence level will increase the width of theconfidence interval and vice versa.Decreasing the variance or decreasing the confidence level will decrease the confidenceinterval and vice versa.Example 1: In the first eight games of this year’s basketball season, Lenny made 25 free throwsin 40 attempts.a. What is p̂ , Lenny’s sample proportion of successes?Let’s quickly check the conditions:1. We have an SRS. Even if it’s not stated in the problem, we can assume it’s an SRS.2. The population is at least 10 times the sample (assume he won’t get hurt and makemany more free throws).3. npˆ 40 25 / 40 25 10 and n(1 pˆ ) 40 1 25 / 40 15 10b. Find and interpret the 90% confidence interval for Lenny’s proportion of free-throw success. 1 confidence level z * qnorm 2 Then pˆ z * pˆ (1 pˆ )nConfidence Interval:Interpretation:Section 7.2 – Confidence Interval for a Proportion2
Example 2: Mars Inc. claims that they produce M&Ms with the following n10%Blue10%A bag of M&Ms was randomly selected from the grocery store shelf, and the color counts were:Brown22Red22Yellow22Orange12Green15Blue15Find the 95% confidence interval for the proportion of yellow M&Ms in that bag.a. What is the proportion of yellow M&Ms in this bag?b. Find and interpret the 95% confidence interval for the proportion of yellow M&Ms in thisbag. 1 confidence level z * qnorm 2 Then pˆ z * pˆ (1 pˆ )nConfidence Interval:Section 7.2 – Confidence Interval for a Proportion3
Sometimes we are asked to find the minimum sample size needed to produce a particular marginof error given a certain confidence level. When working with a one-sample proportion, we canpˆ (1 pˆ )In these problems, we’ll be looking for n souse the formula: Maximum ME z * npˆ (1 pˆ )we could simply plug into: n 2ME / z * If p̂ is unknown, use an estimate of p. If p is unknown, just assume 50, 50, so use p 0.5.Example 3: It is believed that 35% of all voters favor a particular candidate. How large of asimple random sample is required so that the margin of error of the estimate of the percentage ofall voters in favor is no more than 3% at the 95% confidence level?p Max ME 1 confidence level z * qnorm 2 n pˆ (1 pˆ ) ME / z * 2Section 7.2 – Confidence Interval for a Proportion4
Example 4: An oil company is interested in estimating the true proportion of female truckdrivers based in five southern states. A statistician hired by the oil company must determine thesample size needed in order to make the estimate accurate to within 1% of the true proportionwith 89% confidence. What is the minimum number of truck drivers that the statistician shouldsample in these southern states in order to achieve the desired accuracy?p ME 1 confidence level z * qnorm 2 n pˆ (1 pˆ ) ME / z * 2Section 7.2 – Confidence Interval for a Proportion5
Section 7.2 – Confidence Interval for a Proportion 2 Facts about Confidence Intervals For smaller n, the confidence interval becomes wider. For larger n, the confidence interval becomes narrower. Increasing the variance or increasing the confidence level will increase the width of
0.1 0.0 x f(x) Sampling Distribution of the Mean 95% Confidence Interval: n 40 0.4 0.3 0.2 0.1 0.0 x f (x) Sampling Distribution of the Mean 95% Confidence Interval: n 20 When sampling from the same population, using a fixed confidence level, the larger the sample size, n, the narrower the confidence interval.
the population. For a 95% confidence interval with 9 degrees of freedom the t-score is 2.262. Table 1. lists the t-scores for specific degrees of freedom and sizes of confidence interval. For a 95% confidence interval, you would use the
Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 95% Confidence Intervals Summary for Sample Data. According to the output, the AndersonDarling Normality Test has
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Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 95% Confidence Intervals Summary for Life in Hours Figure 4.6: Graphical Summary of TV Component Life Data Calculating Descriptive Stati
Chapter 6: Estimation and Confidence Intervals. . the 95% confidence level interval,. Correct 19 times out of 20 1 time out of 20, by chance, our interval doesn’t contain the parameter (either below or above our interval) . confidence inte
1.051 1.032 1.071 1.034 1.011 1.057 95.8% Group full none partial Overall Mean Lower Bound Upper Bound 95% Confidence Interval for Mean Median Lower Bound Upper Bound Actual Coverage 95% Confidence Interval for Median The confidence interval for the median is constructed without any distribution assumptions. The actual coverage level may be
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