1 / 11

Lesson 8 - 1

Lesson 8 - 1. Distribution of the Sample Mean. Objectives. Understand the concept of a sampling distribution Describe the distribution of the sample mean for sample obtained from normal populations

ciro
Download Presentation

Lesson 8 - 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lesson 8 - 1 Distribution of the Sample Mean

  2. Objectives • Understand the concept of a sampling distribution • Describe the distribution of the sample mean for sample obtained from normal populations • Describe the distribution of the sample mean from samples obtained from a population that is not normal

  3. Vocabulary • Statistical inference – using information from a sample to draw conclusions about a population • Standard error of the mean – standard deviation of the sampling distribution of x-bar

  4. Conclusions regarding the sampling distribution of X-bar • Shape: normally distributed • Center: mean equal to the mean of the population • Spread: standard deviation less than the standard deviation of the population • Law of Large numbers tells us that as n increases the difference between the sample mean, x-bar and the population mean μ approaches zero

  5. Central Limit Theorem X or x-barDistribution Regardless of the shape of the population, the sampling distribution of x-bar becomes approximately normal as the sample size n increases. Caution: only applies to shape and not to the mean or standard deviation x x x x x x x x x x x x x x x x Random Samples Drawn from Population Population Distribution

  6. Summary of Distribution of x

  7. What Happens to Sample Spread If the random variable X has a normal distribution with a mean of 20 and a standard deviation of 12 • If we choose samples of size n = 4, then the sample mean will have a normal distribution with a mean of 20 and a standard deviation of 6 • If we choose samples of size n = 9, then the sample mean will have a normal distribution with a mean of 20 and a standard deviation of 4

  8. a Example 1 The height of all 3-year-old females is approximately normally distributed with μ = 38.72 inches and σ = 3.17 inches. Compute the probability that a simple random sample of size n = 10 results in a sample mean greater than 40 inches. P(x-bar > 40) μ = 38.72 σ = 3.17 n = 10 σx = 3.17 / 10 = 1.00244 x - μ Z = ------------- σx 1.28 = ----------------- 1.00244 40 – 38.72 = ----------------- 1.00244 = 1.277 normalcdf(1.277,E99) = 0.1008 normalcdf(40,E99,38.72,1.002) = 0.1007

  9. a Example 2 We’ve been told that the average weight of giraffes is 2400 pounds with a standard deviation of 300 pounds. We’ve measured 50 giraffes and found that the sample mean was 2600 pounds. Is our data consistent with what we’ve been told? P(x-bar > 2600) μ = 2400 σ = 300 n = 50 σx = 300 / 50 = 42.4264 x - μ Z = ------------- σx 200 = ----------------- 42.4264 2600 – 2400 = ----------------- 42.4264 = 4.714 normalcdf(4.714,E99) = 0.000015 normalcdf(2600,E99,2400,42.4264) = 0.0000001

  10. Summary and Homework • Summary: • The sample mean is a random variable with a distribution called the sampling distribution • If the sample size n is sufficiently large (30 or more is a good rule of thumb), then this distribution is approximately normal • The mean of the sampling distribution is equal to the mean of the population • The standard deviation of the sampling distribution is equal to σ / n • Homework • pg 431 – 433; 3, 4, 6, 7, 12, 13, 22, 29

  11. Homework • 3) standard error of the mean • 4) zero • 6) population is normal • 7) four • 12) μ=64, σ= 18, n=36 μx-bar =64, σx-bar= 18/36 = 18/6 = 3 • 13) μ=64, σ= 18, n=36 μx-bar =64, σx-bar= 18/36 = 18/6 = 3 • 22) μ=81.7, σ= 6.9 • a) P(x < 75) = 0.1658 normalcdf(-E99,75,81.7,6.9) • b) n=5 P(x<75) =0.01496 normalcdf(-E99,75,81.7,6.9/5) • c) n=8 P(x<75) =0.00301 normalcdf(-E99,75,81.7,6.9/8) • d) very unlikely (or unusual) • 29) P(x < 45) = 0.0078 normalcdf(-E99,45,50,16/60)

More Related