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Probability and Statistics – Mrs. Leahy Study Guide – Unit 7 – ESTIMATING Situation 1: Estimating µ when σ is known Conditions that MUST be met: 1. Sample must be random 2. Sample Size > 30 OR population follows a normal distribution 3. σ must be known Step 1: Identify from your problem the following σ = the standard deviation of the population n = the size of the sample 𝑥̅ = the average (mean) of the sample c = the confidence level for the problem Step 2: Identify the critical value zc Use the Level of Confidence Table (you will be provided this table) Step 3: Calculate E, the margin of error (you will be provided this formula) Step 4: Determine the Confidence Interval for the Mean of your population, µ (you will be provided this formula) Step 5: Interpret the Confidence Interval “We are __C%__ confident that the true mean is in the interval ____ to ____.” (or similar statement) Example: At a local high school, a random sample 50 students were measured and their heights (in inches) were recorded. The sample mean height 𝑥̅ = 58 inches. Assume that σ is known to be 5.8 inches. Find a 98% confidence interval for the entire student population at the high school. Give a brief interpretation of the confidence interval in the context of this problem. Conditions met? Situation 2: Estimating µ when σ is unknown Conditions that MUST be met: 1. Sample must be random 2. σ must be unknown Step 1: Identify from your problem the following n = the size of the sample 𝑥̅ = the average (mean) of the sample s = the standard deviation of the sample c = the confidence level for the problem d.f. = n –1 = degrees of freedom Step 2: Identify the critical value tc Use the t-distribution table If d.f. isn’t listed, use the closest, smaller value (you will be provided the entire table) Step 3: Calculate E, the margin of error (you will be provided this formula) Step 4: Determine the Confidence Interval for the Mean of your population, µ (you will be provided this formula) Step 5: Interpret the Confidence Interval “We are __C%__ confident that the true mean is in the interval ____ to ____.” (or similar statement) Example: John is selling vacuum cleaners door to door. He wants to estimate the average amount of sales at the end of each month. A random sample of the last 24 months had a mean of 𝑥̅ = $2500 and a standard deviation of s = $350. Find an 80% confidence interval for the mean amount of sales each month. Give a brief interpretation of the interval in the context of this problem. Conditions met? How much in sales could he expect for 6 months of work? Situation 3: Estimating p for a binomial distribution Conditions that MUST be met: 1. n= fixed number of trials 2. only two outcomes, success & failure 3. 𝑛𝑝̂ > 5 and 𝑛𝑞̂ > 5 𝑞̂ Step 1: Identify from your problem the following n = the number of trials r = the number of successes out of n trials 𝑝̂ = the probability of success in the n trials 𝑞̂ = the probability of failure in the n trials c = the confidence level for the problem Step 2: Identify the critical value zc Use the Level of Confidence Table (you will be provided this table) Step 3: Calculate E, the margin of error (you will be provided this formula) 𝐸 ≈ 𝑧𝑐 √ 𝑝̂𝑞̂ 𝑛 Step 4: Determine the Confidence Interval for the Mean of your population, µ (you will be provided this formula) 𝑝̂ − 𝐸 < 𝑝 < 𝑝̂ + 𝐸 Step 5: Interpret the Confidence Interval “We are __C%__ confident that the true probability of success is in the interval ____ to ____.” (or similar statement) Example: A random sample of 100 students at a local high school found that 75 say they watch TV for more than 2 hours per day. Find a 95% confidence interval for p. Make a brief interpretation of the interval based on the problem. Situation 4: Determining the minimum sample size n Determining the sample size Step 1: Determine using your (Situation 1) formulas/tables/etc. zc = from provided table E = “within E of the mean” (from problem) Step 2: Evaluate for n using one of the following formulas (provided) Sample Size for Mean (µ) Sample Size for Proportion (Probability) p Examples: a) A study is planned to estimate the mean number of hours students at a local high school watch TV. The population standard deviation is assumed to be σ = 5.6 hours. How many students should be included in the sample to be 95% confident that the sample mean 𝑥̅ will be within 1.5 hours of the population mean µ ? b) Let p be the proportion of students who own an iphone or other smartphone. What size sample is needed to be 99% sure that the point estimate 𝑝̂ will be within 0.05 of p ? c) Let p be the proportion of students who are saving money for college. A preliminary study shows that 76% of high school students are saving money for college. How large a sample is needed to be 98% sure that the point estimate 𝑝̂ will be within 0.08 of p ?