normal distribution?

Question

can someone show me how to do this? Show work please?

Use a standard normal distribution table to find the percent of the total area under the standard normal curve between the following z-scores.
Z=0.55 and z=1.3

Answer 1

We know that the total area of the normal distribution from negative infinity to positive infinity is 1 because the normal distribution is a probability density function. However, just 3 standard deviations above and below the mean accounts for about 0.9973002 of the normal distribution.



On most z-tables when we calculate the probability of the z-scores we get the total area from negative infinity to that z-score, or in the other words the area under the normal distribution to the left of the z-score. If we want the area between z-scores it is just the difference of the probabilities for the larger z-score minus that of the smaller z-score.



Adding up areas:

prob(Z = 0.55) + prob(0.55 <= Z <= 1.3) = prob(Z = 1.3),

where prob(0.55 <= Z <= 1.3) is the area between the z-scores 0.55 and 1.3.

Algebraic rearrangement:



prob(0.55 <= Z <= 1.3) = prob(Z = 1.3) - prob(0.55)



= 0.9032 - 0.7088



= 0.1944.



So the percent total area under the standard normal curve between the following z-scores Z = 0.55 and Z = 1.3

explains approximately 19.44% of the total area of the normal distribution from negative infinity to positive infinity which is 100%



See the following table for calculating the areas. The page shows you how to do it.

https://www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/the-normal-curve-39/finding-the-area-under-the-normal-curve-193-2641/



Of course, the statistical programming language R can give us a more exact area between those z-scores which is 0.1943592 ... found by calculating:



> pnorm(1.3) - pnorm(0.55)

[1] 0.1943592

Answer 2

A typical Z-score chart, like the one referenced below, gives the cumulative percentage from the extreme left side of the curve (essentially, Z → -∞), to a given point.

Look at the second chart, the one that says "Positive z-scores," and find the row that starts with "0.5" which represents the percentages for z-scores of 0.50 to 0.59. Find the box in that row that is in the column that starts with "0.05" and you will see "0.7088." This is the total area under the curve from all-the-way-to-the-left up to z= 0.55. Do the same for "1.3" and you should find "0.9032."

You want the area between z=0.55 and z=1.3. If you subtract the area of (ExtremeLeft to 0.55) from (ExtremeLeft to 1.3) you will be left with the area you want. You should get 0.1944.

Answer 3

P(Z<1.3) = 0.9032

P(Z<0.55) = 0.7088

Answer 0.9032-0.7088 = 0.1944

Answer 4

Use a standard normal distribution table

--->Pr(Z<0.55)=0.7088

and Pr(Z<1.3)=0.9032

---> Pr(0.55