The Standard Normal Distribution (Z-Distribution) 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 the transformation 𝑧= 𝑥 − 𝜇 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 Z-transformation, 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. 𝑎= 565 −600 80 b= 710 −600 80 𝑎=−0.438 𝑏=1.38 Example

Another reason z-scores are helpful 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.

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.

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.

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