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Tests and Intervals for a Single Population Proportion

Tests and Intervals for a Single Population Proportion. Example. Population of students. CI: What proportion (percentage) of students abstain from alcohol? HT: Is it more than 20%?. How likely is it that the sample proportion would be as extreme as 0.234 if p = 0.20?. Sample of

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Tests and Intervals for a Single Population Proportion

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  1. Tests and Intervals for a Single Population Proportion

  2. Example Population of students CI: What proportion (percentage) of students abstain from alcohol? HT: Is it more than 20%? How likely is it that the sample proportion would be as extreme as 0.234 if p = 0.20? Sample of 175 students

  3. “Theory” for One Proportion • Binary variable: person either has trait (“1”) or does not have trait (“0”). • Then, since proportion is number with trait divided by total number, a proportion is basically an average of 0’s and 1’s. That is, p = (0 + 0 + 1 + … + 1 + 0 + 1)/n. • Central Limit Theorem: For large samples, sample proportions will be at least approximately normally distributed.

  4. As long as sample is “large” as defined by: • More than 5 people are in sample with trait, and • More than 5 people are in sample without trait. If n = sample size, and p0 = value of proportion in null hypothesis Inference for One Proportion Confidence Interval Hypothesis Test

  5. Example: CI for One Proportion • What proportion of students abstain from alcohol? • 0.23 ± 1.96(0.23)(0.77)/175=0.23 ± 0.06 • We can be 95% confident that between 17% and 29% of all students abstain from alcohol.

  6. Example: HT for One Proportion • Is the proportion who abstain more than 0.20? • H0: p = 0.20 versus HA: p > 0.20 • Z = (0.234 - 0.20)/ (0.20)(0.80)/175 = 1.13 • P-value = P(Z > 1.13) = 0.129 • There is not enough evidence to conclude that the proportion is more than 0.20.

  7. In Minitab with Raw Data • Select Stat. Select Basic Statistics. Select 1 Proportion… • In box labeled “Samples in Columns,” specify the binary variable of interest. • Under Options…, specify the confidence level, the null and alternative hypotheses. If a large sample, check box in front of “Use test and interval based on normal dist’n.” • Select OK.

  8. In Minitab with Summarized Data • Select Stat. Select Basic Statistics. Select 1 Proportion… • Click on button in front of “Summarized Data.” In box labeled “Number of trials,” put n, the number in your sample. In box labeled “Number of successes,” put the number with the trait of interest. • Under Options…, do the same as previous. • Select OK.

  9. Example: Minitab Outputfor One Proportion Test and Confidence Interval for One Proportion Test of p = 0.2 vs p > 0.2 Success = 1 Variable X N Sample p 95.0 % CI Z P-Value abstain 41 175 0.234286 (0.172, 0.297) 1.13 0.128

  10. 95% margin of error is If p-hat is ½, then 95% margin of error is Trick for quick estimate of 95% margin of error The largest the margin of error can be is when p-hat is ½.

  11. Example • 100 students sampled • 25% of those sampled love almonds • 95% margin of error is no larger than 1/100 = 1/10 = 0.10 • Confidence interval is 0.25 ± 0.10. • We can be 95% confident that between 15% and 35% of all students love almonds.

  12. As always… • P-values and confidence intervals are only accurate if the assumptions are met. • Check to make sure you have a large enough sample.

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