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8.1 Estimating μ when σ is Known. Access for population data is either expensive or impossible to attain so we have to estimate the population parameters Assumptions : We have a Simple Random Sample (SRS) of size n from our population.
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8.1 Estimating μ when σ is Known • Access for population data is either expensive or impossible to attain so we have to estimate the population parameters • Assumptions: • We have a Simple Random Sample (SRS) of size nfrom our population. • The value ofσ (the population standard deviation) is known. • If x distribution normal, our methods work with any sample size of n. • If unknown distribution, then a sample size of n ≥ _____ is required. If it is distinctly skewed or not mound shaped a sample of 50 or 100 may be necessary.
8.1 Estimating μ when σ is Known • Point estimate of a population parameter is an estimate of the parameter using a single number. _____ is the point estimate for μ When using x as a point estimate for μ, the margin of error is | x - μ | Also, we cannot be certain that our point estimate is correct but we can have a certain percent of confidence that it is correct.
8.1 Estimating μ when σ is Known • For Confidence level c, the critical value zc is the number such that the area under the standard normal curve is between –zc and zc ,equals c. • Find 99% confidence z value. • Find 95% confidence z value
8.1 Estimating μ when σ is Known • Interpretation of confidence interval “We are 95% confident that mean math SAT score of all high school seniors is between 452 and 470” • Confidence level c • Gives probability that the interval will capture the true parameter value in repeated samples • Interpretation: “95% of all possible random samples of size 500 will result in a confidence interval that includes the true mean math SAT score of all high school seniors”
8.1 Estimating μ when σ is Known • margin of error is | x - μ | = zcσ = E n So x- E μ x + E or x – zcσ μ x + zcσ n n
8.1Example 1:A jogger jogs 2 miles/dayTime is recorded for 90 daysσ = 1.8 min, x =15.6 minFind a 95% confidence interval for μ , the average time required for a 2-mile jog over the past year.
8.1 Estimating μ when σ is Known • To find confidence interval • Find Error E = zc σ/ n • Subtract from x on low side and add to x on high side. • Write range.
Example 2: • A sociologist is studying the length of courtship before marriage in a rural district of Kyoto, Japan. A random sample of 56 middle-income families was interviewed. It was found that the average length of courtship was 3.4 years. If the population standard deviation was 1.2 years, find an 85% confidence interval for the length of courtship for the population of all middle-income families in this district.
8.1 Estimating μ when σ is Known • Sample size for estimating the mean . Sometimes we want an know how many samples need to be taken to limit the error. n = zcσ2 E
Example 3 • Light fixtures Assembly • Suppose we need the mean time μ to assemble a switch. • In 45 observations, s = 78 seconds. • Find the number of additional observations needed to be 95% sure that xbar and mu will differ by no more than 15 seconds.
8.2 Inference for mean when unknown • Use t-distribution with s (standard deviation of sample) as point estimate for. • Standard error: SEx = s/ n s = std dev.of sample n = number of trials d.f. = n - 1 (degrees of freedom) • t-distribution: • similar in shape to normal distribution • spread is greater (height lower in the middle; more probability in tails) • As d.f (degrees of freedom) increases (n-1) • t- distribution becomes more normal • Also s approaches as n increases
8.2 Estimating µ when σ is unknown • Finding t confidence level • Find tc for 0.95 confidence level when the sample size is 17. • Find tc for 0.75 confidence level when the sample size is 15. • Find tc for .75 confidence level when the sample size is 2 • Find tc for 0.99 confidence level when the sample size is 300.
8.2 Inference for mean when unknown • Margin of Error = t* s/ n • t-confidence interval • x bar – t* s/ sqrt (n) x bar+ t* s/sqrt(n) • t – value for t(n-1) for specified confidence level c • Conditions: • SRS • Normal population or close (Look for extreme skewness or outliers, or bimodel)
8.2 Inference for mean when unknown • T procedures can be used if: n 40 except with large outlier in data n 15 okay except for strong skewness or outliers N 15 data needs to be close to normal
8.2 Example 1 • Suppose an archaeologist discovers only seven fossil skeletons from a previously unknown species of miniature horse. The shoulder heights (cm) are • 45.3 47.1 44.2 46.8 46.5 45.5 47.6 • Find xbar and s, Assume population is approximately normal. • Find the degrees of freedom • Find tc for a confidence level of .99 • Find E (error) • Find the confidence interval • State the confidence level related to this problem.
8.2 Example 2 • A company has a new process for manufacturing large artificial sapphires. In a trial run, 37 sapphires are produced. The mean weight for these 37 gems is • xbar = 6.75carats, and the sample standard deviation is s = .33 carat. • Find the degrees of freedom • Find tc for a confidence level of .95 • Find E (error) • Find the confidence interval • State the confidence level related to this problem.
