Slide 1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1 n Learning Objectives –Identify the properties of the normal distribution and normal curve –Identify the characteristics of the standard normal curve –Understand examples of normally distributed data –Read z-score tables and find areas under the normal curve –Find the z-score given the area under the normal curve –Compute proportions –Check whether data follow a normal distribution The normal distribution Chapter S7

Slide 2 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 2 n Learning Objectives continued... –Understand and apply the Central Limit Theorem –Solve business problems that can be represented by a normal distribution –Calculate estimates and their standard errors –Calculate confidence intervals for the population mean The normal distribution Chapter S7

Slide 3 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 3 Normal distribution frequencies of observations frequency polygon smooth bell-shaped curve a normal distribution. When the frequencies of observations for a large population result in a frequency polygon that follows the pattern of a smooth bell-shaped curve that population is said to have a normal distribution.

Slide 4 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 4 Normal distribution n bell-shaped n symmetrical about the mean n total area under curve = 1 one standard deviation n approximately 68% of distribution is within one standard deviation of the mean two standard deviations n approximately 95% of distribution is within two standard deviations of the mean 3 standard deviations n approximately 99.7% of distribution is within 3 standard deviations of the mean n Mean = Median = Mode

Slide 5 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 5 Normal curves same mean but different standard deviation

Slide 6 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 6 Standard score ( z -score) The z -score of a measurement is defined as the number of standard deviations the measurement is away from the mean. The formula is:

Slide 7 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 7 Standard score ( z -score) μ If a distribution has a mean of μ and a standard deviation of σ the corresponding z -score of an observation is:

Slide 8 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 8 Conversion to raw scores Z -score is calculated to determine the appropriate areas under any normal curve. –To convert raw score of x (from a distribution with a mean μ and standard deviation σ to a z -score, subtract the mean from x and divide by the standard deviation. –To convert a z -score to a raw score x, multiply the z -score by the standard deviation and add this product to the mean. In equation form:

Slide 9 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 9 The Central Limit Theorem means approximately normally distributed If random samples of size n are selected from a population with a mean μ and a standard deviation σ, the means of the samples are approximately normally distributed with a mean μ and a standard deviation,even if the population itself is not normally distributed, provided that n is not too small. The approximation becomes more and more accurate as the sample size n is increased.

Slide 10 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 10 Confidence intervals n Point estimates –a single estimate of an unknown population mean can be obtained from a random sample –different random samples give different values of the mean point estimate –a single estimate is referred to as a point estimate –accuracy depends on: variability of data in the population size of the random sample

Slide 11 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 11 Standard error of mean n Standard error of the mean point estimate of the mean n Standard error of the mean provides the precise measure of accuracy of a point estimate of the mean Where: σ = population standard deviation = size of random sample

Slide 12 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 12 Confidence intervals confidence interval n A range of values in which a particular value may lie is a confidence interval. level of confidence. n The probability that a particular value lies within this interval is called a level of confidence.