10.1 Estimating with Confidence

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1 10.1 Estimating with Confidence

2 Point Estimator and point estimate A point estimator s a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our best guess at the value of an unknown parameter. When we use a sample mean (xbar) for the population mean, it is an example of a point estimate.

4 Exp The admission directory of Big City University has a novel idea. He proposed using the IQ scores of current students as a marketing tool. The university agrees to provide him with enough money to administer IQ tests to 50 students. So the director gives the IQ test to an SRS of 50 of the university s 5000 freshman. The mean IQ score for the sample is xbar=112. What can the director say about the mean score of the population of all 5000 freshman? Is the mean IQ score of all Big City University freshman 112? Let s look at the sampling distribution. We can calculate the Sampling distribution standard deviation for this sample of n=50. The Central Limit Thereom tells us that the sampling distribution of xbar is approximately normal.

9 WH Freeman TPS4e Confidence Interval Applet Rossman Chance Confidence Interval Applet

15 Interpretation Confidence level (C): For example let C=95%. In repeated samples of this size, 95% of the invervals obtained will contain the true (parameter name) Confidence Interval: I am 95% confident that interval from to contains the actual, (parameter in context)

16 Conditions for Constructing a Confidence Interval for mu (standard deviation is known) It is appropriate to construct a confidence interval when: S SRS provided the data from the population of interest I independent observations, when sampling without replacement, the sample size must be 10% or less than the Population size. N sampling distribution of xbar is approximately Normal

17 Why do we check these conditions??? S - the random conditions helps ensure that the statistic is unbiased, i.e., xbar or phat I independent condition ensures we are using the correct formula for the standard deviation of the statistic N the normality condition ensures we are using the correct critical value (z*)

18 Exp 1 This problem refers to a random sample of 35 teenagers who averaged 7.3 hours of sleep per night. Assume the population standard deviation is 1.8 hours. Calculate a 95% confidence interval for the mean. Step 1: Problem state the problem identifying parameters. Identify the population of interest and the parameter you want to draw conclusions about. Find estimate of mu for the mean number of hours slept by teenagers.

19 Step 2: Conditions check them S Simple random sample n = 35 I number of hours slept per teenager is independent N Since n >= 30, the sampling distribution of the mean is approximately normal. Step 3: Calculations estimate +/- margin or error = xbar +/- (critical value)(standard deviation) = 7.3 +/- (1.96)(1.8/sqrt(35)) = 7.3 +/-.596 (6.704,7.896)

20 Step 4: Conclusion in context We are 95% confident that the interval between hours and hours contains the actual mean number of hours slept by teenagers.

21 Example 10.5 Video Screen Tension p630 A manufacturer of high-resolution video terminals must control the tension on the mesh of the wires that lies behind the surface of the viewing screen. Too much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mv). Some variation is inherent in the production process. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is 43 mv (sigma). Here are the tension readings from an SRS of 20 screens from a single day s production Construct and interpret a 90% confidence interval for the tension of all screens from a single day s production.

22 Step 1: Parameter Identify the population of interest and the parameter you want to draw conclusions about. Find estimate of mu for the mean tension of screens for the day s production. Step 2: Check conditions S Simple random sample n = 20 I production of screens is independent N Check for sampling distribution of xbar to be approximately normal. The sample size is too small to assume an approximately Nnormal distribution (n is not greater than or equal 30)

23 Check data: create boxplot of data and check for outliers or strong skewness Create Normality Plot to see if data is approximately normally distributed This problem sketch bloxplot The distribution has no outliers or strong skewness.

24 Check Normality Plot on calculator. [Stat plot: last plot available] If it is approximately linear, then the distribution is approximately normal. Sketch not necessary. This problem: The Normality plot is approximately linear. It is reasonable to assume the distribution is approximately Normal.

25 **NOTE: check mathshepherd.com under 10.1 for Assessing Normality.pdf Step 3: Calculations estimate +/- margin or error = xbar +/- (critical value)(standard deviation) = x ± z *( σ n ) = ±1.645( ) = / (290.5, ) Step 4: Interpretation We are 90% confident that the interval between mv and mv contains the actual mean tension in the entire batch of terminals produced that day.

26 How do changes in Confidence Level affect the margin of error? a) Find confidence interval for 80% confidence level. b) Find confidence interval for 99% confidence level. How do changes in sample size affect the margin of error? a) Find the margin of error for 90% confidence level n = 50. b) Find the margin of error for 90% confidence level n = 100. How do changes in standard deviation affect the margin of error? a) Find margin of error for 90% confidence level with standard deviation = 50 mg/dl. b) Find margin of error for 90% confidence level with standard deviation = 75 mg/dl.

27 We want High Confidence (method almost always gives correct answers) with Small Margin of Error pin down parameter quite Precisely. As n increases - margin of error decreases

28 As Confidence Level decreases - margin of error decreases

29 As sigma (standard deviation) decreases margin of error decreases Estimate +/- (critical value)[sigma/sqrt(n)] **Note: It is the size of the sample that determines the Margin of error. The size of the population does not influence the sample size we need as long as the population size is much larger than the sample.

30 CAUTION: Data must be an SRS No correct method for inference from data haphazardly collected with bias of unknown size Outliers can distort results The shape of the distribution matters You must know the standard deviation of the population Next section addresses what to do when standard deviation is not known.

31 Exp Researchers would like to estimate the mean cholesterol level of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 mg/dl of the true value of mu at a 95% confidence level. A previous study suggests that the standard deviation of cholesterol level is 5 mg/dl. Obtaining monkeys is time-consuming and expensive, so they want to know the minimum number of monkeys needed to generate a satisfactory estimate.

32 m >= z*[standard deviation] Find z* - critical value 1 area invnorm ( ) = (1.96)( 5 n ) n = 9.8 n >= à Always round up to next whole number n = 97 The minimum number of monkeys required to produce an estimate within 1 mg/dl is 97.

33 Common mistakes on AP Exam Expressing confidence interval in terms of probability. (The probability of the true mean being in the confidence interval is either 0 or 1) Memorize the magic words. Confusing Confidence Level with Confidence Interval each time a confidence interval is created you are to interpret it. You only interpret the Confidence Level when specifically asked to do so. Estimating the Confidence Interval with first checking the conditions (SIN) Estimating Confidence Intervals without including their interpretation.

34 2000 FR Q2 (Cave Footprints) Anthropologists have discovered a prehistoric cave dwelling that contains a large number of adult human footprints. To study the size of the adults who used the cave dwelling, they randomly selected 20 of the footprints from the population of all footprints in the cave and measured the length of those footprints. Some statistics resulting from this random sample are as follows. Sample size 20 Minimum 15.2 cm Mean 24.8 cm Q cm Standard Dev 7.5 cm Median 21.5 cm Q3 30.cm Maximum 37.0 cm The anthropologists would like to construct a 95% confidence interval for the mean foot length of the adults who used the cave dwelling.

35 a) What assumptions are necessary in order for this confidence interval to be appropriate? b) Discuss whether each of the assumptions listed in your response to (a) appears to be satisfied in this situation.

36 2002A FR Q1 (Einstein/Newton) In 1915 Einstein s theory predicted that the curvature of space, denoted by y was 1, while Newtonian theory predicted it was 0. Since 1915 scientists have repeatedly found estimates of y using various methods and procedures. Each method has a margin of error. The figure below displays (estimate +/- margin of error) from each of 21 experiments.

37 a) Based on the display above, describe how the precision of the estimates of y has changed over time. b) Write a few sentences describing the strength of evidence the experiments provide for the claim from Newtonian theory that y = 0. Your response must include justification based on the display. c) Write a few sentences describing the strength of evidence the experiments provide for the claim from Einstein s theory that y = 1. You response must include justification based on the display.

Chapter 8: Estimating with Confidence

Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE The One-Sample z Interval for a Population Mean In Section 8.1, we estimated the