Use Table 4. Appendix 3, to find the following probabilities for a standard normal random variab

Use Table 4. Appendix 3, to find the following probabilities...

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. It serves as a reference distribution for statistical tests and probability calculations, and its properties allow us to standardize normal random variables for comparisons and table look-ups.

Z-Score and Standardization

A Z-score represents the number of standard deviations a data point is from the mean. Standardization converts any normal random variable to a standard normal variable, enabling the use of universal tools like the Z-table to determine probabilities and percentiles.

Cumulative Distribution Function (CDF) for the Normal Distribution

The CDF for the normal distribution gives the probability that a standard normal random variable is less than or equal to a specified value. This function is essential for determining probabilities corresponding to intervals under the curve by finding the difference between two cumulative values.

Z-Score Table (Standard Normal Table)

A Z-score table is a reference tool used to look up the cumulative probabilities associated with Z-scores. It facilitates the process of finding the areas under the standard normal curve between specified values and is a key resource in statistical analysis of normally distributed data.

Symmetry of the Normal Distribution

The normal distribution is symmetric about its mean, meaning that probabilities for positive and negative deviations of equal magnitude are identical. This symmetry is often visualized using horizontal axes on probability tools, reinforcing that the distribution’s behavior is mirror-like on either side of the mean.

Transcript

00:01 Welcome to numerid.

00:02 In the current problem, we will find a couple of probabilities.

00:06 To start with, we will have to find the probability of 0 less than equals to z less than equals to 1 .2 equals to how much.

00:18 So, if i bring this over here and try to explain you what i am trying to say, this will be 0 and see this is 1 so 1 .2 will be somewhere over here so we have to find how much area is this so if you see that value beyond 0 is always 0 .5 and value beyond this 1 .2 can be found out from the staple so instead of writing this we can write this to be probability of z greater than equals to 0 minus probability of z greater than equals to 1 .2 and we have this value to be 0 .5 minus now 1 .2 if you see 1 .2 is over here which is 0 .11151 so 0 .1151 and if i subtract that that would give me 948 0 .3 .49.

01:32 So this would be the answer to this problem.

01:36 Next will be, next up we have another problem.

01:43 So i will use up this space only to explain.

01:51 So let me just drag a little bit up here so that i get this place for writing.

01:59 Next problem is probability minus 0 .9 less than said less than 0 .0.

02:07 What is minus 0 .9? of course this is 0 and this will be minus 1.

02:14 So minus 0 .9 will be this point.

02:16 I want to find what area is this minus 0 .9.

02:22 So this would be nothing but the area this this this this this this this in between area we can also think of in terms of this how we did.

02:34 So whatever is less than this area is the same as more than this area, correct? this is 0 .9 positive.

02:44 So we can again write probability area less than 0, probability z less than equals to 0 minus probability z less than equals to minus 0 .9, which can be substituted by probability z greater than 0 .9.

03:09 It's the same tail left right.

03:11 0 .9 is this.

03:13 So this value will be 0 .1841.

03:16 That is 0 .5 minus 0 .1841 which would result in 0 .1 first 9 and then 5 and then 1 and then 3 .0 .359.

03:30 This is the answer of part b.

03:33 So let me quickly clean the screen.

03:41 And clean this up as well this.

03:48 Now moving to problem number c, part c, probability 0 .3 less, oops, it looks like 6 .3, 0 .3 less than equal to z less than equal to 1 .56 will be equals to.

04:10 Now see, this is where zero lies.

04:12 So, 0 .3 would be somewhere over here.

04:17 This 1 .5, this is 1 .56 will be over here.

04:21 How much area is there in between this two? so we can clearly write probability z greater than equals to 0 .3.

04:30 That will be the whole area minus this smaller white area, this area.

04:36 So minus probability z greater than equals to 1 .56.

04:41 What is the probability of 0 .3? greater than this.

04:45 0 .3821 minus.

04:52 What is the probability of this? 1 .56.

04:56 So 1 .5 is over here and 1 .56 would be this value.

05:00 So 0 .0594.

05:05 0 .0594.