Refer to Exercise 4.68 . If students possessing a GPA less than 1.9 are dropped from college, wh

step 1 Welcome to Numerid. In the current problem, again, we are facing a variable X, which is the GPs

Refer to Exercise 4.68 . If students possessing a GPA less...

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical around its mean, forming a characteristic bell-shaped curve. It is widely used in statistics to model real-world phenomena such as test scores, measurement errors, and other natural characteristics, making it a fundamental concept for probability analysis.

Z-Score Standardization

The z-score is a statistical measure that represents how many standard deviations a data point is from the mean of a distribution. Standardizing data using z-scores allows for the comparison of different values within a normal distribution, simplifying the process of determining probabilities associated with a specific threshold.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a particular value. In the context of a normal distribution, the CDF is used to calculate the proportion of data that falls below a given threshold, which is essential for identifying percentages of observations, such as students below a certain GPA.

Tail Probability

Tail probability refers to the probability of values falling in the extreme ends (tails) of a probability distribution. It is especially important in decision making when thresholds are applied, such as dropping students with a GPA below a predetermined cut-off, by quantifying how much of the distribution lies beyond this boundary.

Transcript

00:01 Welcome to numerid.

00:02 In the current problem, again, we are facing a variable x, which is the gps of a large population of students.

00:11 And also it is mentioned that x is following a normal distribution where mu is 2 .4 and sigma is 0 .8.

00:20 Now, we will call a student is dropping off if the gpa is less than 1 .9.

00:28 So if we want to know what is the point.

00:30 Percentage of drop off then we would have to find the probability of that event so this is our event in which we are interested so how do we proceed we always proceed by standardizing the variable so here is our format to standardize subtracting the mean and dividing by the sd so of course this we call as per applet probabilities this is z and this two will be equal and this two will be equal.

01:07 All the steps will be equal because of applet probabilities because this is just a change of scale and origin.

01:13 It does not change the distribution and the alongside probabilities.

01:17 So this is 1 .9 minus 2 .4 divided by 0 .8 .8.

01:24 So that would be probability of z.

01:27 Less than minus 0 .5 divided by 0 .8, which is probability z less than minus 0 .625.

01:42 Now can you think of any standard result that can be used here to over here that will ease, make our life easy? can we write this? think, think, think, think.

01:57 I will explain.

01:58 Why it is can we do this so the formula in question is probability of z less than minus a is equal to probability of z greater than plus a think of minus one and plus one can't think okay let you bring the diagram for you but if i just insert standard standard normal variables here, if you see, this is the probability below minus 1, correct? this shaded region, correct? and what is the probability of z greater than plus 1? which is this...

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