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Math 011 – CHAPTER 7 Estimates and Sample Sizes 7-2 Estimating a Population Proportion Notation: GOALS: DEFINITION A point estimate A Confidence Interval (CI) A confidence level Example 1: (Interpreting a Confidence Interval) – Based on the sample data of 1007 adults polled, with 85% of them knowing what Twitter is, we can find a confidence interval: The 95% confidence interval estimate of the population proportion p is 0.828 < p < 0.872 Correct Interpretation: Incorrect Interpretation(s): Note: Example 2 (Finding a critical value zα/2) Find the critical value zα/2 corresponding to a 90% confidence level. Common Critical Values Level α zα/2 DEFINITION Confidence Interval for Estimating a Population Proportion p Requirements: 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied. 3. There are at least 5 successes and at least 5 failures. Confidence Interval: ˆ E p p ˆ E where p Also expressed as or E z 2 ˆ ˆ pq n ˆ ( p p̂ Round the confidence interval limits for p to three significant digits. Example (Constructing a Confidence Interval) – In a poll of 1007 randomly selected US adults, 85% of the respondents know what Twitter is. a. Find the margin of error that corresponds to a 95% confidence level. b. Find the 95% confidence interval estimate of the population proportion p. c. Based on the results, can we safely conclude that more than 75% of adults know what Twitter is? d. Write a brief summary that accurately describes the results and include all the relevant information. When analyzing polls: Finding the Sample Size n Required to Estimate a Population Proportion Requirement: The sample must be a simple random sample of independent sample units. When an estimate is p known: When no estimate is p known: Rounding: If the computed sample size n is not a whole number, round the value UP to the next larger whole number. Example 4 (Computing a Sample Size) – Many companies are interested in knowing the percentage of adults who buy clothing online. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points? a. Use the recent result from the Census Bureau: 66% of adults buy clothing online. b. Assume that we have no point estimate. Caution: Finding the Point Estimate P and Error E from a Confidence Interval Example 5: Find the point estimate and margin of error of the statement: “95% confident that the population proportion is from 60% to 85%”. 7-3 Estimating a Population Mean Confidence Interval for Estimating a Population Mean µ with σ Not Known Requirements: 1. The sample is a simple random sample. 2. Either or both of these conditions is satisfied. a. The population must be normally distributed b. n > 30 Confidence Interval: x E x E where (Use df = n – 1) where tα/2 = critical value separating an area of α/2 in the right tail of the student t distribution, and df = the number of degrees of freedom is the sample size minus 1. Also expressed as X ±𝐸 or Round-off Rules: Important Properties: E Example 1: (Finding a Critical t Value) a. A sample of size 40 is a simple random sample obtained from a normally distributed population. Find the critical value tα/2 corresponding to a 95% confidence level. b. A sample size of 25 is a simple random sample obtained from a normally distributed population. Find the critical value tα/2 corresponding to a 90% confidence level. Example 2 (Constructing a Confidence Interval) – In a test of weight loss programs, 40 adults used the Atkins weight loss program. After 12 months, their mean weight loss was found to be 2.1lbs, with a standard deviation of 4.8lbs. Construct a 90% confidence interval estimate of the mean weight loss for all such subjects assuming the population is normally distributed. Finding a Point Estimate and Margin of Error E from a Confidence Interval Finding the Sample Size n Required to Estimate a Population Mean Requirement: The sample must be a simple random sample. Use Formula 7-4: Rounding: If the computed sample size n is not a whole number, round the value UP to the next larger whole number. Confidence Interval for Estimating a Population Mean µ with σ Known Requirements: 1. The sample is a simple random sample. 2. Either or both of these conditions is satisfied: a. b. Confidence Interval: x E x E where E z 2 n where zα/2 = critical z score separating an area of α/2 in the right tail of the student normal distribution. Also expressed as X ±𝐸 or Example 3 (Confidence Interval Estimate of µ with known σ) – Twelve highway speeds were measured from southbound traffic on 1-280 near Cupertino, California. The simple random sample has mean of 60.7mi/h. Construct a 95% confidence interval estimate of the population mean by assuming that σ is known to be 4.1. Construct the confidence interval. Section 7-4 Estimating a Population Standard deviation or Variance n 1 s 2 2 2 Chi-Square Distribution: Properties of the Chi-Square Distribution: 1. 2. 3. Note: In table A-4, each critical value of χ2 in the body of the table corresponds to an area given at the top row of the table, and each area is a cumulative area to the RIGHT of the critical value. Example 1: (Finding Critical Values of χ2) – A simple random sample of 22 IQ scores is obtained. Find the left and right critical values corresponding to a confidence level of 95%. Confidence Interval for Estimating a Population Standard Deviation or Variance Requirements: 1. The sample is a simple random sample. 2. The population must have normally distributed values (even if the sample is large). n 1s Confidence Interval for σ2: 2 2 R n 1s Confidence Interval for σ: Round-Off Rules: 2 R 2 2 n 1s 2 L2 n 1s 2 2 L Example 2 (Constructing a Confidence Interval) – IQ scores for subject in three different lead exposure groups were recorded. The 22 full IQ scores for the group with medium exposure to lead have a standard deviation of 14.3. Consider the sample to be a simple random sample and construct a 95% confidence interval estimate of σ. Determining Sample Size: Refer to Table 7-2 to determine the sample size depending on the desired margin of error and level of confidence. σ2 σ To be 95% confident that s is within 1% 5% 10% 20% 30% To be 99% confident that s is within 1% 5% 10% 20% 30% Of the value σ, the sample size n should be at last 19,205 768 192 48 21 Of the value σ, the sample size n should be at last 33,218 1,336 336 85 38 To be 95% confident that s2 is within 1% 5% 10% 20% 30% To be 99% confident that s2 is within 1% 5% 10% 20% 30% Of the value σ2, the sample size n should be at last 77,208 3,149 806 211 98 Of the value σ2, the sample size n should be at last 133,449 5,458 1,402 369 172 Example: We want to estimate the standard deviation of σ all IQ scores of people with exposure to lead. How large should be sample if: a. we want to be 95% confident that our estimate is within 10% of the true value of σ? b. we want to be 95% confident that our estimate is within 5% of the true value of σ? c. we want to be 99% confident that our estimate is within 1% of the true value of σ? Example: Consider the following weights of post-1983 pennies: 3.1582, 3.0406, 3.0762, 3.0398, 3.1043, 3.1274, 3.0775, 3.1038, 3.0586, 3.0603, 3.0502, 3.1028, 3.0522 Assume the sample is a simple random sample obtained from a population with a normal distribution. Construct a 98% confidence interval estimate of the standard deviation of the weights of all post-1983 pennies.