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Week 6 Vocabulary Summary Section 6.1– Confidence Intervals for the Mean (Large Samples) Point Estimate: is a single value estimate for a population parameter. The most unbiased point estimate of the populations mean µ (mu) is the mean x xbar (sample mean) Interval Estimate: is an interval, or range of values, used to estimate a population parameter. To find an interval estimate, + - the statistic by a margin of error Level of Confidence: c is the probability that the interval estimate contains the population parameter. The Central Limit Theorem States: ( page 272) 1. If samples of size n, where n >= 30 are drawn from a population with a mean of µ and a standard deviation of sigma., then the sampling distribution of the sample means approximates a normal distribution. The greater the sample size, the better the approximation. 2. If the populations itself is normally distributed, the sampling distribution or sample means is normally distributed for any sample size n. Sample Mean: x Standard Deviation: x n The standard deviation of the sampling distribution of the sample means x is called the standard error of the mean. Translation: If n >= 30, use the assumed mean and standard deviation for calculations For any size sample n, use modifications if the population is normally distributed. Critical values: are values that separate sample statistics that are probable from sample statistics that are improbable, or unusual. Represent by zc. On one side are values that are probable, on the other improbable. Most common used: Level of Confidence zc 90% 1.645 95% 1.96 99% 2.575 Sampling error: is the difference between the point estimate and the actual parameter value. When µ is estimated the sampling error is x xbar - µ mu Margin of Error (maximum error of estimate or error tolerance) : Represent by E. Given level of confidence C, E is the greatest possible distance between the point estimate and the value of the parameter it is estimating. E zc * x zc * n Round-Off Rule: Round off the confidence interval for the population mean to the same number of decimal places given for the sample mean STEPS Construction the Confidence Intervals for the Population Mean IF sigma IS KNOWN The steps 1. Find the sample statistics n and xbar x x n 2. Specify sigma 3. Find the critical value zc that corresponds to the given level of confidence 4. Find the margin of error E zc * n 5. Find the left and right endpoints and form the confidence interval. x E left endpoint x E right endpoint Interval x E x E STEPS Construction the Confidence Intervals for the Population Mean IF sigma IS NOT KNOWN The steps 1. Find the sample statistics n and xbar x x n 2. If n >= 30, find the sample standard deviation s estimate for sigma 3. Find the critical value ( x x) n 1 2 and use it as an zc that corresponds to the given level of confidence 4. Find the margin of error E zc * n 5. Find the left and right endpoints and form the confidence interval x E left endpoint x E right endpoint Interval x E x E STEPS In reverse: find the minimum sample size n, given c-confidence and a margin of error E If sigma is unknown, estimate it using s, provided you have a preliminary sample with at least 30 members Formula is derived from the Margin of error formula above. E zc * n E * n zc * n zc * E z * n c E 2 Section 6.2– Confidence Intervals for the Mean (Small Samples n < 30) t-distribution: If the distribution of the random variable x is approximately normal, then x t s n Properties of a t-distribution: 1. Bell-shaped and symmetric about the mean 2. Family of curves, each determined by a parameter called “degrees of freedom” 3. Total area under a t-curve = 1 (100%) 4. The mean, mode, median, are centered on the curve at 0. 5. As the “degrees of freedom” increases, the t-distribution approaches the normal distribution. After 30df the t-distribution is very close to the standard normal z-distribution df – “degrees of freedom”: defined as the number of free choices left after a sample statistic is calculated. d . f . n 1 See page 325 STEPS Construction the Confidence Intervals for the Mean: t-distribution The steps 1. x , and s ( x x) Find the sample statistics n, x n 2 n 1 2. Find the “degrees of freedom” d.f.= n - 1 3. Find tc from table #5 4. Find the margin of error E tc * s n 5. Find the left and right endpoints and form the confidence interval x E left endpoint x E right endpoint Interval x E x E Section 6.3– Confidence Intervals for Population Proportion p Population Proportion: is the proportion of success (concepts learned about the Binomial distribution) Point Estimate: for the population proportion p. x , x is the number of successes in the sample and n is number in the sample. n qˆ 1 pˆ , is the proportion of failures. pˆ c-confidence Interval for the population proportion p: pˆ E p pˆ E , where E zc ˆˆ pq n The probability that the confidence interval contains p is c STEPS Construction the Confidence Intervals for the Population Proportion The steps 1. Find the sample statistics n and x. 2. Find the point estimate pˆ x n 3. Verify that the sampling distribution of p̂ can be approximated by the normal distribution 4. Find the critical value zc that corresponds to the given level of confidence 5. Find the margin of error E zc ˆˆ pq n 6. Find the left and right endpoints and form the confidence interval. p̂ E left endpoint p̂ E right endpoint Interval pˆ E p pˆ E Find the minimum sample size to estimate p WITH OR WITHOUT preliminary estimate available The steps x , if one cannot be found use 0.50 for p-hat n 2. Verify that the sampling distribution of p̂ can be approximated by the normal 1. Find the point estimate pˆ distribution 3. Find the critical value zc that corresponds to the given level of confidence z ˆ ˆ c 4. Find the sample size n pq E 2