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Week3&amp;4_Zs (1)

# Week3&amp;4_Zs (1) - Week 3&4 Z tables and the Sampling...

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Week 3&4: Z tables and the Sampling Distribution of ¯ X

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Week 3&4: Z tables and the Sampling Distribution of ¯ X The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N ( 0 , 1 2 ) . The distribution of any other normal random variable, X N ( μ, σ 2 ) , can be converted to a Z = X - μ σ . Probabilities for these variables are areas under the curve, but since we don’t use calculus in the this course, we can use software or a Z table to find probabilities. The random variable, Z , is continuous which means the probabilty at any exact point is always 0. Thus, we will find probabilities for ranges of values. 2 / 37
Week 3&4: Z tables and the Sampling Distribution of ¯ X First, some general characteristics of the Z distribution. The area under the entire curve is 1 since it represents all possible values. Because it is symmetric, the mean = the median, so the area under the curve to the left of 0 is 0.5 (as is the area to the right). We say, “The probability that Z is less than 0 is 0.5.” This is written as P ( Z < 0 ) = 0 . 5. Again, since Z is continuous, P ( Z 0 ) = P ( Z < 0 ) = 0 . 5. 3 / 37

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Week 3&4: Z tables and the Sampling Distribution of ¯ X We will use the Z table found on the Stat30X webpage - http://www.stat.tamu.edu/stat30x/zttables.php Notice that the only entry on both pages of the table is z = 0.00 and the probability is 0.5000. The rows of the table are the z -scores with the columns indicating the 2 nd decimal. The body of the table contains the probabilitiesis to the left of any particular z -score = z.zz. For example, the P ( Z < 0 . 00 ) = 0 . 5000 and P ( Z < 0 . 07 ) = 0 . 5279. 4 / 37
Week 3&4: Z tables and the Sampling Distribution of ¯ X Examples of reading the table: P ( Z < 1 . 25 ) = 0 . 8944 P ( Z < 0 . 50 ) = 0 . 6915 5 / 37

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Week 3&4: Z tables and the Sampling Distribution of ¯ X P ( Z < - 0 . 75 ) = 0 . 2266 P ( Z < - 2 . 01 ) = 0 . 0222 6 / 37
Week 3&4: Z tables and the Sampling Distribution of ¯ X The Z -table only gives probabilities to the left of a value. If we want to get probabilities to the right we use the complement rule, P ( Z > z ) = 1 - P ( Z < z ) . P ( Z > 1 . 25 ) = 1 - 0 . 8944 = 0 . 1056 P ( Z > 0 . 50 ) = 1 - 0 . 6915 = 0 . 3085 7 / 37

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Week 3&4: Z tables and the Sampling Distribution of ¯ X P ( Z > - 0 . 75 ) = 1 - 0 . 2266 = 0 . 7734 P ( Z > - 2 . 01 ) = 1 - 0 . 0222 = 0 . 9778 8 / 37
Week 3&4: Z tables and the Sampling Distribution of ¯ X To find probabilities between two numbers, find the larger area (using the larger value) first and then subtract the smaller area. Remember, a probability can never be negative, so check your work! P ( - 2 . 01 < Z < 2 . 01 ) = P ( Z < 2 . 01 ) - P ( Z < - 2 . 01 ) = 0 . 9778 - 0 . 0222 = 0 . 9556 9 / 37

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Week 3&4: Z tables and the Sampling Distribution of ¯ X Now suppose we have a non-standard normal, X N ( μ, σ 2 ) , and we want to know the probability that X is less than some value. We must first convert the X to a Z and then use the probabilities from the Z -table. Recall that if X N ( μ, σ 2 ) , then Z = X - μ σ N ( 0 , 1 2 ) so P ( X < x ) = P ( Z < x - μ σ ) Beware : P ( X > x ) 6 = 1 - P ( X < x ) if X is not centered at 0. You must convert to a Z before using the complement rule.
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Week3&amp;4_Zs (1) - Week 3&4 Z tables and the Sampling...

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