The population standard deviation σ of a discrete random variable ‚ Measure how close a random variable tends to be the population mean μ‚ so you must understand μ before you understand σ If you have a random variable like a bet at a casino or and investment then the standard deviation σ measure the risk‚ if there is a lot of risk then the standard deviation is high The formulas for standard deviation are given below but you should look at the examples first Population mean Population variance
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Sample sizes and confidence intervals for proportions Chong Chun Wie Ext: 2768 ChongChunWie@imu.edu.my Content • Sampling distribution of sample means (SDSM) • Normality Test • Estimating a population mean: σ known • Estimating a population mean: σ unknown • Standard deviation of proportion • Confidence interval of proportion • Hypothesis testing with proportion Population and Sample Samples Populations Sampling distribution of sample means (SDSM) Sampling distribution • Example – Select
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distribution with a standard deviation of $40‚000. a. If we select a random sample of 50 households‚ what is the standard error of the mean? b. What is the expected shape of the distribution of the sample mean? c. What is the likelihood of selecting a sample with a mean of at least $112‚000? d. What is the likelihood of selecting a sample with a mean of more than $100‚000? e. Find the likelihood of selecting a sample with a mean of more than $100‚000 but less than $112‚000. a) The standard error of the mean
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below. The tabulation is in the decreasing order of cost. 1. If the graduate applies to all 10 companies‚ what is the probability that she will get at least one offer? 2. If she can apply to only one company‚ base on cost and success probability criteria alone‚ should she apply to company 5? Why or why not? 3. If she applies to companies 2‚5‚8‚ and 9‚ what is the total cost? What is the probability that she will get at least one offer? 4. If she wants to be at least 75% confident of getting
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Practice Problems for Confidence Intervals X ± Z σ/n ; X ± t s/n ; p ± Z (p(1-p)/n) 1. A press release issued by our university claims that West Chester students study at least as much as the national average for students at four year universities. Across the nation‚ 73 percent of all students at four year universities study at least four hours per week. Seventy percent of one hundred randomly selected West Chester students surveyed claimed to study more than four hours per week.
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that malfunction and produce excess radiation. A random sample of 40 machines is taken and 12 of the machines malfunction. The problem is to compute the 95% confidence interval on π‚ the proportion that malfunction in the population. Solution: The value of p is 12/40 = 0.30. The estimated value of σp is = 0.072. A z table can be used to determine that the z for a 95% confidence interval is 1.96. The limits of the confidence interval are therefore: Lower limit = .30 - (1.96)(0.072) = .16
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included in y f. Represents the simplest measure of spread (or variability) g. Indicates the most frequent observation in a frequency distribution h. Represents a theoretical family of distributions that may have any mean or any standard deviation i. Indicates how widely the out around the measures of central tendency j. Measure an event over time B. Match the following data display tools with their descriptions. (Descriptions may be used more than once or not at all.) A horizontal or vertical
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histogram of the 1990 returns. (ii) Produce a histogram of the 1998 returns. (iii) Find the mean‚ median‚ range and standard deviation for the 1990 returns. Annual Returns % (1990) Mean 12.91865979 Median 11.38 Standard Deviation 9.297513067 Range 75.01 (iv) Repeat part (iii) for the 1998 returns. Annual Returns % (1998) Mean 6.355463918 Median 5.4 Standard Deviation 5.170830853 Range 42.76 (v) Which was the better year for investors? • 1990 was the better year for investors in
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Wentworth Medical Center in upstate New York investigated the relationship between geographic location‚ health status ( healthy or one or more comorbidities)‚ and depression. Random samples of 20 healthy individuals were selected from three geographic locations: Florida‚ New York‚ and North Carolina. Then‚ each was given a standardized test to measure depression (higher test scores indicate higher levels of depression). Similarly‚ random samples of 20 individuals with one or more comorbidities (arthritis
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Confidence Intervals have numerous applications for professional activities. Confidence Intervals have a wide use in defining the outcome of a particular question. The use of confidence levels are used commonly in Health‚ Business‚ Politics and Engineering venues. There are three examples that will be recognized as having real world applications regarding confidence intervals. An Empirical Test of the Black-Scholes (BS) Option pricing model exhibited the use of a confidence interval approach
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