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CH 24C The Standard Normal Distribution (Z-Distribution) • Apples from a grower’s crop were normally distributed with mean of 173 grams and standard deviation 34 grams. Apples weighing less than 130 grams were too small to sell. • Use notation to describe the information above. X ~ N(173, 342) • Find the proportion of apples from this crop that were too small to sell. P(X<130) = 10.3% • The top 15% of the apples are sent to Trader Joes. What is the minimum weight for an apple to be sent to Trader Joes? k = 208g Warm Up • The volume of a cool drink in a bottle filled by a machine is normally distributed with mean 503 mL and standard deviation 0.5 mL. 1% of the bottles are rejected because they are under-filled, and 2% are rejected because they are over-filled; otherwise they are kept for retail. What range of volumes is in the bottles that are kept? Between 502 mL and 504 mL Application Example • The standard normal distribution or Z-distribution has a mean of 0 and a variance of 1, hence a standard deviation of 1. • Every normal X distribution can be transformed into the standard normal distribution or Z-distribution using 𝑥−𝜇 the transformation 𝑧 = . 𝜎 • The notation for the Z-distribution is Z ~ N(0,1). • The z-score is the number of standard deviations a data value is above (or below) the mean. • Z-scores are useful when comparing two populations with different parameters 𝜇 and 𝜎. 24C • z-scores are useful when comparing two populations with different 𝝁 and 𝝈. • Before graphing calculators and computer packages, it was impossible to calculate probabilities for a general normal distribution. 𝑁 𝜇, 𝜎 2 . • Instead all data was transformed using the Ztransformation, and the standard normal distribution table was consulted for the required probability values. 24C Significance of the Z-distribution Example 1 • The standardized verbal scores of students entering a large university are normally distributed with a mean of 600 and a standard deviation of 80. • The probability that the verbal score of a student lies between 565 and 710 is represented by the shaded area in the following diagram. This diagram represents the standard normal curve. • Write down the values of a and b. 565 −600 𝑎= 80 𝑎 = −0.438 710 −600 b= 80 𝑏 = 1.38 a 0 Example b • For some questions we MUST convert to z-scores to solve. • We always need to convert to z-scores if we are trying to find an unknown mean or standard deviation. Another reason z-scores are helpful Example: • The weights of baby boys follow a normal distribution with a mean of 3.3 kg. It is known that 85% of these babies have a weight less than 3.8 kg. Find σ. X ~ N (3.3, 2 ) P( X 3.8) .85 3.8 3.3 PZ .85 3.8 3.3 1.04 0.5 1.04 .482 Example: • The heights of certain flowers follow a normal distribution. It is known that 20% of these flowers have a height less than 3 cm and 10% have a height greater than 8 cm. Find u and σ. 3 PZ 0.2 3u .8416 2.35cm u 4.98 cm 8 PZ 0.9 8u 1.2816 IB Example: • The speeds of cars at a certain point on a straight road are normally distributed with mean u and standard deviation σ. 15 % of the cars travelled at speeds greater than 90 km h–1 and 12 % of them at speeds less than 40 km h–1. Find u and σ. 90 40 PZ PZ 0.85 0.12 90 u 40 u 1.03643 1.17499 22.62 km h 1 u 66.56 km h 1 Last Example : Technology