STT 315 Exam 3 Formula sheet
STT 315 Exam 3 Formula sheet STT 315
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This 2 page Study Guide was uploaded by Jordyn on Friday April 1, 2016. The Study Guide belongs to STT 315 at Michigan State University taught by Eroglu in Fall 2015. Since its upload, it has received 342 views. For similar materials see Intro to Business Statistics in Business at Michigan State University.
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Date Created: 04/01/16
Parameter is a numerical descriptive measure of population o Some common parameters are population mean (μ)), population standard deviation (σ) and population proportion (p) [population proportion is fixed] A sample statistic (or just statistic) is a numerical descriptive measure of a sample. It is calculated from the observations in the sample o Some common statistics are sample mean (xx), sample standard deviation (s), sample proportion (p) [sample proportipn is sample of people from selected population] A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the parameter A statistic is said to be an unbiased estimator of the parameter if the mean of the sampling distribution of a statistic is equal to the parameter If the mean of the sampling distribution is not equal to the parameter the statistic is said to be a biased estimator of the parameter The expected value xx: μ = Exxx) = μ, xx is an unbiased estimator of μ The standard deviation of xx: σ = σxxx) = σ/√n Central Limit Theorem (CLT): If n is sufficiently large, the sampling distribution of xx will be approximately a normal distribution with mean μ = μxxnd standard deviation σ = σ/√n, Wxxcan consider the sample size “large enough” if n ≥ 30 Then sample proportion of success is: p = x/n p The standard deviation of p σ = p(pp = √[(p(1-p))/n], n is considered “large enough” if both np≥15 and n(1-p) ≥ 15 Statistical inference means that we are making a conclusion about the population parameter based on the statistic we calculated from a sample There are two types of inference: 1. Estimation: we estimate an unknown parameter value 2. Hypothesis test: we verify of a claim about the parameter(s) is true The unknown population parameter (e.g. mean or proportion) that we are interested in estimating is called target parameter There are two times of estimation: 1. Point estimation: provides a single value based on observations from one sample 2. A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter (xx)—[point estimation does not give any info about ME] Interval estimation provides a range of values, based on observations from one sample An interval estimator (or confidence interval) is a formula that tell us how to use the sample data to calculate an interval that estimates the target parameter with certain level of confidence The probability distribution of z-scores: z-statistic: Z = (xx-μ)/(σ/√n) If the value of the population std. deviation σ is known, the large sample 95% CI for μ is: xx±1.96(σ/√n); xx ± ME = statistic ± ME (xx-zα/2σ/√n), xx+z *α/2√n)), we compute z α/2= invNorm(1-α/2, 0, 1) Large sample 100(1-α)% CI for μ: [use Z-interval] o Larger the σ, larger the ME: Larger the CI, larger the ME: Larger the n, smaller the ME EX: CI of μ is (1.9818, 1.9982): Width=1.9982-1.9818 =0.0164… therefore ME= ½ * width = 0.0082 The estimated standard deviation of xx is known as standard error of xx and SE=s/√n If σ is unknown the margin of error of large sample CI for μ is: ME = z *(s/√n) α/2ritical value * SE Small sample 100(1-α)% CI for μ: [use T-interval with unknown σ and Z-interval with known σ] If the sample size is small (n<30) the CLT cannot be applied Compared to standard normal all t-distributions have heavier tail area, but lower peak The t-statistic: T: = (xx-μ)/(s/√n) follows t-distribution with (n-1) degrees of freedom (df) [larger the df less variability there is] To compute P(a < T < b): tcdf(upper, lower, df) To compute area of T distribution: InvT(1-(α/2), df) Optimal sample size for estimating μ Always round up for optimal sample size of n Large sample 100(1-α)% CI for p: [use 1-PropZInt] The point estimator for p is the sample proportion of successes p=x/n p μ pp and σ =√ppp*(1-p))/n] ME= z α/2 √[(p*(1-p))/n]: CI = p±pz α/2* √[(p*(1-p))/n]: n is sufficiently large if both np ≥p15 and n(1- p)≥15p Optimal sample size for estimating p if no information about p is given, use p=0.5 Test of hypothesis Hypothesis: a claim/conjecture about some parameter of the population distribution o H o the null hypothesis: represents the hypothesis that will be accepted unless the data provides convincing evidence that it is false. Usually represents some claim about the population parameter that the researcher wants to test o H a the alternative hypothesis: represents the hypothesis that will be accepted only if the data provides convincing evidence of its truth. Usually represents values of pop parameter that researchers want to gather evidence to support Type I error: we find evidence to reject H but 0t turns out that H is tru0 Type II error: we do not find evidence to reject H but i0 turns out to be false Reality H is true H is true 0 a Type I error is Reject H0 Type I error (False Correct decision worse than Type Decision positive) II Do not reject Correct decision Type II error (false H 0 negative) Example: a drug company tests the null hypothesis that a new drug does not work better than a placebo, the Type I error would say the new drug works better than the placebo, but it really does not Level of significance (usually denoted by α) is the maximum permissible probability of the Type I error If “p-value is less than level of significance” i.e. if p-value < α we reject H 0 Test statistic: is a sample statistic, computed from information provided in the sample that the researcher uses to decide between the null and alternative hypothesis Tests for population proportion Z= Ha: p > p0-- then p-value = P(standard normal Alternative hypothesis Reject rv ≥ Z) H o Ha: p < p0--then p-value = P(standard normal H a p > p 0 Z > z α rv ≤ Z) H a p < p 0 Z < -z α H a p ≠ p 0 |Z| > z α/2
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