For a normal distribution what standard score Zscore has 40

For a normal distribution, what standard score (Z-score) has 40% of the distribution above it? Find the closest value.Give your answer to 2 decimal places

Solution

A) Since 40% are in the tail, and there are two tails, this is splitted into two regions, each containing 20%. In the left tail, this means that 20% are to the left of the particular z score that divides the data. In the right tail, this means that 20% are to the right of the particular z score that divides the data. More importantly, 80% are to the left of the right z score. The z score table gives you the area to the left, so Inside the table, find the exact value, if not the closest for either 0.2 (20%) or 0.8 (80%). Chances are, you won\'t be able to find 0.2, but you\'ll see 0.8 or a value close to 0.8.

The closest value is 0.7995. Look to the left column of the row containing this value. You\'ll see 0.8. This is the ones and tenth digit of the z score. Now look at the upper row of the column that contains 0.7995. You\'ll see 0.04. This is the hundredth digit of the z score. Putting them together, we see that the z score = 0.84. This is the right side z score.

Since normal distribution is symmetrical, the left z score is simply opposite of the right z score, which is z = -0.84.

If you have graphing calculator, you can find the exact value. Press [2nd], [vars (Distr)], and select invNorm(. Input the area to the left of the z score. Unlike the table, you can use either 0.2 or 0.8. You\'ll get z = +_ 0.8416212335 which is more accurate than the one we found using the table.