Template for CI, Ho testing and sample size for Proportions (Fall 2023 Includes problems and solutio

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Virginia Commonwealth University *

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Statistics

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Nov 24, 2024

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xlsx

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70

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Confidence Interval for Propo Given binomial random variable with Y success •M ean (Y)or Expected value of Y= 𝑛∗? •Variance (Y) = 𝑛∗? ∗(1−?) •Standard deviation (Y) = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐? = √(𝑛∗? ∗(1 The Central Limit Theorem applies. Recall that variable can be approximated by a normal rand For a given sample of n trials and Y successes, •The sample proportion ? ̂ = 𝑌/𝑛 •The sampling distribution of the sample prop distributed. (? ̂−?)/√((? ̂(1−? ̂))/𝑛) is approximately a 95% CI for estimation of ? = ? ̂ ± Z 𝛼⁄2 * √((? This is predicated on the binomial random approximated by a random normal variabl when the following conditions are met: 1. n ? ̂ -5 ≥ 0 2. n ? ̂ +5 ≤ n If E(Y) is too close to 0 or n, the normal distribu because there is too much area to the left of 0
ortion ses in n trials 1−?) ) a binomial random dom variable. portion ? ̂ is normally standard normal ? ̂(1−? ̂))/𝑛) m variable being le. This is only valid ution cannot be used 0 or right of n.
Estim n (sample size) 125 Number that meet the outcome studied 84 -hat ? 0.672 Does it meet conditions? Value n -5>/0 *? 79 n +5</n *? 89 Sample Calculations Point estimate hat) (? 0.672 Standard Error= sqrt hat(1 hat)/n] [? -? 0.04 Confidence interval Confidence interval (enter proportion) 0.95 Z multiple/Critical Z 1.96 95%Confidence Interval Lower Limit 0.5897 Upper Limit 0.7543 Confidence Interval Length 0.1646 Hypothesis Testing 0.6 Z statistic (Zsample) 1.643 Expected proportion (p o )
Test (One Sample) Ha p≠0.6 p<0.6 p>0.6 Test (Choose from drop down list) Right-tailed test Critical Z value 1.645 Z statistic (from your sampling) 1.643 p-value 0.0502 Decision cannot reject Ho Two sided test( H 0 : p=p o , H a : p≠p o ) Left tailed test (H o : p≥p o , H a : p<p o ) Right tailed test (H o : p≤p o , H a : p>p o )
mating Proportions (One Sample) Enter Enter Meets Condition? yes yes Enter l Enter To use this template, only en will autopopulate. Rememb conditions to be able to cons proportion. To confirm that the template C12: 125 C13: 84 C25: .95 If done correctly, the 95% CI Test Statistic Z= ((𝑝 ̂−𝑝 0 ))/√((𝑝 0 (1−𝑝 0 ))/𝑛)
95% Conf Interval Z Critical Value Lower Bound Upper Bound p-value +/- 1.960 0.5897 0.7543 0.1003 -1.645 - ∞ 0.7411 0.9498 1.645 0.6029 + ∞ 0.0502
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