Calculating Probabilities for Any Normal Variable Sketch the normal curve Shade the region of interest and mark the delimiting x-values Compute the z-scores for the delimiting x-values found in step 2 Use the standard normal table to obtain the probabilities 2/16/2019

Find the Z-Score Suppose that you are given a probability for a normally distributed random variable. In order to find the (z) value corresponding to that probability: Sketch the normal curve associated with the variable Shade the region of interest Use the table to obtain the z-score Convert the z-score to an x-value using the following formula: x = μ + σz 2/16/2019

MBA 510 February 19 Spring 2013 Dr. Tonya Balan 2/16/2019

Sampling Error The distribution of a sample statistic is called the sampling distribution. Sampling error is the error resulting from using a sample statistic to estimate a population parameter. These characteristics help us to understand how accurate an estimate is likely to be. 2/16/2019

Sampling Distribution of the Mean For a variable X and a given sample size n, the distribution of the variable 𝑥 is called the sampling distribution of the mean. Distribution of the variable 𝑥 Distribution of all possible sample means 2/16/2019

Example Consider a population that consists of the five starters on a basketball team. Further, suppose that X is a random variable that represents the heights of the players. The data are given below: Suppose that we take a random sample of size n=2 and use that sample to estimate the average height of the players. Player A B C D E Height 76 78 79 81 86 2/16/2019

Example Player A B C D E Height 76 78 79 81 86 Sample Heights 𝒙 A B 78 77.0 A C 79 77.5 A D 81 78.5 A E 86 81.0 B C B D 79.5 B E 82.0 C D 80.0 C E 82.5 D E 81 86 83.5 2/16/2019

Example (larger samples) Player A B C D E Height 76 78 79 81 86 Sample Heights 𝒙 A B C D 78 79 81 78.50 A B C E 78 79 86 79.75 A B D E 78 81 86 80.25 A C D E 79 81 86 80.50 B C D E 81.00 2/16/2019

Sampling Distribution of the Mean The possible sample means cluster closer around the population mean as the sample size increases. The larger the sample size, the smaller the sampling error of the mean. For a sample of size n, the mean of the random variable 𝒙 equals the mean of the variable under consideration. For a sample of size n, the standard deviation of 𝒙 equals the standard deviation of X divided by the square root of the sample size. 2/16/2019

Sampling Distribution of the Mean If X is a normally distributed random variable with mean μ and standard deviation σ, then the sample mean 𝑥 has a normal distribution with mean μ and standard deviation 𝜎 𝑛 Central Limit Theorem: Suppose that X is a random variable with mean μ and standard deviation σ. Then, for large samples (n>30), from an population, the sample mean 𝑥 will follow an approximately normal distribution with mean μ and standard deviation 𝜎 𝑛 2/16/2019

Example A normal population has mean 200 and standard deviation 100. Consider samples of size n=50. Find the mean of the sampling distribution of 𝑥 Find the standard deviation of the sampling distribution of 𝑥 Find P(195≤ 𝑥 ≤ 205) Find P( 𝑥 ≥ 210) 2/16/2019

Example At a city high school, past records indicate that the MSAT scores for students have a mean of 510 and a standard deviation of 90. One hundred students in the high school are to take the test. What is the probability that their mean score will be More than 530? Less than 500? Between 495 and 515 2/16/2019

Confidence Intervals A point estimate of a parameter is the value of a single statistic that is used to estimate a population parameter. A confidence interval consists of a point estimate and a margin of error along with a specified confidence level for the interval. 2/16/2019

Example Illustrate confidence interval concepts graphically Weiss p 448 2/16/2019

Interpretation Confidence intervals enable you to make statements such as: “I’m 95% confident that this interval contains the true population mean.” What you can NOT say: “There is a 95% probability that the population mean will be in this interval.” 2/16/2019

CI for One Population Mean When σ Is Known A (1 - 𝛼)% confidence interval for the mean of a normal population with known standard deviation σ is given by: 𝑥 − 𝑧 𝛼/2 𝜎 𝑛 , 𝑥 + 𝑧 𝛼/2 𝜎 𝑛 Note: The higher the confidence level, the wider the interval. The larger the sample size, the smaller the interval. 2/16/2019

Critical Values α α/2 𝒛 𝜶/𝟐 Confidence Level 0.2 0.1 1.28 80% 0.05 1.645 90% 0.025 1.96 95% 0.01 0.005 2.575 99% 2/16/2019

Example Income in a particular city is known to be normally distributed with a standard deviation of $6325. Suppose that a city assessor randomly selects 40 households and finds that the average income for the sample is $29,400. Find a 95% confidence interval for the mean income and interpret. A similar study in the previous year showed a mean income of $25,300. Based on this information, would the assessor conclude that the mean income has increased over the previous estimate? 2/16/2019