Identify two primary reasons for studying the normal distribution. It is often appropriate to use the normal distribution as the distribution of a variable and the normal distribution is frequently employed in inferential statistics. Define the following terms. Normally distributed variable. A variable is said to be normally distributed if its distribution has the shape of a normal curve. Normally distributed population if a variable of a population is normally distributed and is the only variable under consideration then we say that we have normally distributed population. Parameters for a normal curve The parameters for a normal curve are the corresponding mean and standard deviation. Answer true or false to each of the following statements. Give reasons for you answers. Two variables having the same mean and standard deviation have the same distribution. False. Two normally distributed variables having the same mean and standard deviation have the same distribution True. A normal distribution is completely determined by is mean and standard deviation. Explain the relationship between percentages for a normally distributed variable and areas under the corresponding normal curve. They are the same when the areas are expressed as percentages. Identify the distribution of the standardized version of a normally distributed variable. Standard normal distribution. Answer true of false to each of the following statements. Two normal distributions having the same mean are centered at the same place, regardless of the relationship between their standard deviation. True. Two normal distributions having the same standard deviation have the same shape, regardless of the relationship between their means. True. Consider the normal curves having the following parameters: μ = 1.5 and σ = 3; μ = 1.5 and σ = 6.2; μ = -2.7 σ = 3; μ = 0 and σ = 1. Which curve has the largest spread? The second curve. Which curves are centered at the same place? The first and second curves. Which curves have the same shape? The first and third curves Which curve is centered farthest to the left? The third curve. Which curve is the standard normal curve? The fourth curve. What key fact permits us to determine percentages for a normally distributed variable by first converting to z-scored and then determining the corresponding area under the standard normal curve? Key Fact which states that the standardized version of a normally distributed variable has the standard normal distribution. Explain in words how Table II is used to determine the area under the standard normal curve that lies to the left of a specified z-score. Read the area directly from the table. To the right of a specified z-score. Subtract the table area from 1. Between two specified z-scores. Subtract the smaller table area form the larger. Explain in words how Table II is used to determine the z-score having a specified area to its lest under the standard normal curve. Locate the table entry that is closest to the specified area and read the corresponding z-score. Right under the standard normal curve. Locate the table entry that is closest to 1 minus the specified area and read the corresponding z-score. What does the symbol z stand for? The z-score having area to its right under the standard normal curve. State the 68.26-95.44-99,74 rule. From property 1 of the 68,26-95,44-99,74 rule, 68.26% of all people have IQs within one standard deviation to either side of the mean. One standard deviation below the mean is 100 – 16 = 84; one standard deviation above the mean is 100 + 16 = 116. So l8.26% of all people have IQs between 84 and 116. Roughly speaking what are the normal scores corresponding to a sample of observations? They are the observations we would expect to get for a sample of the same size for a variable having the standard normal distribution. If we observe the values of a normally distributed variable for a sample, then a normal probability plot should be roughly linear. Sketch the normal curve with parameters. μ = -1 and σ = 2. μ = 3 and σ = 2 μ = -1 and σ = 0.5.

For the standard normal curve, find the z-scores having area 0.30 to its left. –0.52, having area 0.10 to its right 1.28 z0.025, z0.05, z.0.1 and z0.005. 1.96; 1.645; 2.33; 2.575 That divide the area under the curve into a middle 0.99 area and two outside 0.005 areas. +2.575