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Confidence Intervals and Sample Size Estimates • Point Estimate: A specific numerical value estimate of a parameter. The best point estimate of the population mean is the sample mean. • Example: Want to estimate the age of teachers in SHS. All teachers are surveyed and answer is 38.6 years. This is a point estimate. • Interval Estimate: An interval or a range of values used to estimate a parameter. This estimate may or may not contain the value of the parameter being estimated. • Preferred due to the fact that the sample mean, for the most part, is somewhat different from population mean due to sampling errors. • Example: Avg. Age might be 38.1 < 𝜇 < 39.1 which is 38.6 ± 0.5 years. Properties of Good Estimators • Estimator must be an unbiased estimator. The expected value or mean of the estimates obtained from samples is equal to the parameter being estimated. • Estimator must be consistent. As sample size increases, the value of the estimator approaches the value of the parameter estimated. • Estimator must be relatively efficient. Of all the statistics that can be used to measure a parameter, the relatively efficient estimator has the smallest variance. Confidence Intervals • The probability of being correct can be assigned before an interval estimate is made. • For example: May want to be 90%, 95%, or even 99% sure that the interval contains the true population mean. The larger your confidence gets the larger your interval must be. • Confidence Interval: A specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate. • Confidence level: The probability that the interval estimate will contain the parameter. Confidence Interval Formula for a specific ∝ Z sub alpha over 2 • ∝ : Represents the total area under both tails of the standard normal distribution curve. • ∝ ∕ 2: Represents the area in each one of the tails. • Relationship between ∝ and confidence level is that the confidence level is the percentage equivalent to the decimal value of 1 - ∝, and vice versa. • The second term of our CI Formula is called the maximum error of estimates. • Max error of estimates: The max difference between the point estimate of a parameter and the actual value of the parameter. Steps to find Z sub alpha over 2. • Step 1: Subtract decimal value of percent from 1. This creates your ∝. • Step 2: Take ∝ and divide by 2. This creates area of each tail. • Step 3: Subtract area from 0.5. • Step 4: Find the corresponding area from Step 3 on our Table E chart. • Step 5: Use that z-value in our CI formula. CI Examples • Example 1: Dr. Purnell wishes to find avg. age of teachers in the district. The standard deviation is known to be 4 years. A sample of 55 teachers had an average age of 32.5 years. Find the 95% confidence interval of the population mean. • Example 2: Avg. annual wind speed in Kitty Hawk NC is 15.6 mph. If a sample of 90 days was used to determine the average, find the 99% confidence interval of the mean. Assume standard deviation to be 2.3 mph. Determining Sample Size • Sample size determination is closely related to statistical estimation. • The formula is derived from the Maximum Error of Estimate formula. • All answers should be rounded up if there is any fractional or decimal portion in our answer. • Formula: n = 𝑍 𝑠𝑢𝑏 𝑎𝑙𝑝ℎ𝑎 𝑜𝑣𝑒𝑟 2 ● 𝜎 𝐸 ² Sample Size Examples • Example 1: Researcher is interested in estimating avg. salary of garbage men in a large town. He wants to be 95% sure that estimate is correct. If standard deviation is $950 how large a sample is needed to get the desired info. and to be accurate within $150? • Example 2: A nurse wants to estimate birth weights of babies. How large must sample be if she desires to be 90% confident that the true mean is within 8 ounces of the sample? Standard deviation is known to be 7 ounces. The T-Distribution Similarities to Normal Dist. • Bell shaped. • Symmetrical about the mean. • The mean, mode, and median are equal to 0 and located at the center. • Never touches x-axis. Differences to Normal Dist. • The variance is greater than 1. • It is actually a family of curves based on the concept of degrees of freedom, which is related to sample size. • As sample size increase it approaches the standard normal dist. curve. d.f. = Degrees of Freedom • d.f. : The number of values that are free to vary after a sample statistic has been computed. • They tell the researcher what curve to use when a distribution consists of a family of curves. • Example: Lets say mean of 10 values is 60. This means that 9 out of 10 values are free to vary. Once they have been selected the last value must be a specific number to get a sum of 600 since 600 ÷ 10 = 60. Hence d.f. = n – 1. Using Table F • Need to find the correct value of t sub alpha over 2. • Step 1: Find the correct d.f along the left hand side. ( d.f. = n – 1 ) • Step 2: Find correct confidence level on top. • Step 3: Intersection becomes our answer. • Step 4: Answer will be used in CI formula on next slide. • Do not need to worry about “one tail” or “two tails”. Confidence Interval Formula for when 𝝈 is unknown and n < 30 T-Distribution Examples • Example 1: For a group of 20 students taking a final exam the mean heart rate was 96 beats per minute. Standard deviation was 5. Find the 95% confidence interval of the true mean. • Example 2: A sample of 12 food servers showed an avg. weekly income of $340.40 with a standard deviation of $11. Find the 98% confidence interval of the true mean. Confidence Interval for a proportion • As with means, the statistician, given the sample population, tries to estimate the population proportion. • An interval estimate can be used for a proportion. • The formula is given below: The Symbols for Proportion Notation • “p hat” = X ÷ n, Where X = number of sample units that possess the characteristics of interest and n = sample size. • “q hat” = 1 – “p hat” Rules for using CI for a proportion • #1: n ● p and n ● q must be greater than or equal to 5. Just like binomial check. • #2: Round off to 3 decimal places. CI Example for Proportions • Example 1: In a recent study of 100 people, 78 said that they were satisfied with their current home. Find the 90% confidence interval of the true proportion of individuals who are satisfied with their current home. • Example 2: A nutritionist found that in a survey of 60 families, 32% said they ate apples at least once a week. Find the 95% confidence interval of the true proportion of families who eat apples at least once per week. Minimum sample size for Interval Estimate of a Population Proportion. • It is necessary to round up to obtain a whole number answer. No fractional or decimal answers allowed. • Formula: • If no p “hat” is known must use 0.5 for both p “hat” and q “hat”. Minimum Examples for Pop. Proportion • Example 1: A researcher wishes to estimate, with 98% confidence , the number of people who own an iphone. A previous study shows that 42% of those interviewed had an iphone. The researcher wishes to be accurate within 3% of the true proportion. Find the minimum sample size. • Example 2: The same researcher wishes to estimate the proportion of people who also own an Ipad. She wants to be 95% confident and accurate within 7% of the true proportion. Find the minimum sample size. CI for Variances and Standard Deviation • Variances and standard deviations are just as important as means. • Example: The variance and standard deviation of the medication in a certain prescription plays an important role in making sure the patients gets the proper dosage. • Due to fact that they are both rather important confidence intervals are necessary. • To calculate these intervals a new distribution is needed called the chi-square distribution. Chi-Square Distribution • Similar to the t-distribution in the fact that it too is a family of curves based on d.f. • Symbol for chi-square is 𝜒². (Pronounced “ki”) • Chi-square variable can not be negative and distributions are positively skewed. At roughly 100 d.f. the distribution becomes somewhat symmetrical. Chi-Square Distribution How to read chi-square table. • There are two different values that are going to be used in the formulas for variance and standard deviations. Need to find those 2 numbers first. • Step 1: Need to find ∝ first by subtracting 1- CI. • Step 2: Divide answer above by 2. Use that ∝ ÷ 2 answer and match it to the d.f. Their intersection creates 𝜒²right. • Step 3: Take answer from ∝ ÷ 2 and subtract it from 1. Use that number and the d.f. intersection to create 𝜒²left. • Step 4: Answers from steps 2 and 3 will be used to find confidence intervals for variances and standard deviations. Chi-Square Distribution Example • Find the values of 𝜒²right and 𝜒²left for a 95% confidence interval when n = 18. • Step 1: Need to find ∝ first by subtracting 1- CI. • Step 2: Divide answer above by 2. Use that ∝ ÷ 2 answer and match it to the d.f. This creates 𝜒²right. • Step 3: Take answer from ∝ ÷ 2 and subtract it from 1. Use that number and the d.f. intersection. This creates 𝜒²left. Formulas for CI for Variances and Standard Deviations • CI for Variance: • CI for Standard Deviation: • Remember s = sample standard deviation and s² = sample variance. Problem could give us either so if deviation is given need to square it. If variance is given plug directly in. CI Variance and Standard Deviation Examples • Find the 99% CI for the variance and standard deviation of the weights of 5 gallon containers of paint if a sample of 14 containers has a standard deviation of 1.2 pounds. Assume the variable is normally distributed. • Find the 90% CI for the variance and standard deviation for the lifetime of batteries if a sample of 25 batteries has a standard deviation of 2.1 months. Assume the variable is normally distributed.