Scores on an examination are assumed to be normally...

Scores on an examination are assumed to be normally distributed with mean 78 and variance 36 a. What is the probability that a person taking the examination scores higher than $72 ?$ b. Suppose that students scoring in the top $10 \%$ of this distribution are to receive an A grade. What is the minimum score a student must achieve to earn an A grade? c. What must be the cutoff point for passing the examination if the examiner wants only the top $28.1 \%$ of all scores to be passing? d. Approximately what proportion of students have scores 5 or more points above the score that cuts off the lowest $25 \% ?$ e. Answer parts $(\mathrm{a}),(\mathrm{b}),(\mathrm{c}),$ and $(\mathrm{d}),$ using the applet Normal Tail Areas and Quantiles. f. If it is known that a student's score exceeds $72,$ what is the probability that his or her score exceeds 84?

Scores on an examination are assumed to be normally distributed with mean 78 and variance 36
a. What is the probability that a person taking the examination scores higher than $72 ?$
b. Suppose that students scoring in the top $10 \%$ of this distribution are to receive an A grade. What is the minimum score a student must achieve to earn an A grade?
c. What must be the cutoff point for passing the examination if the examiner wants only the top $28.1 \%$ of all scores to be passing?
d. Approximately what proportion of students have scores 5 or more points above the score that cuts off the lowest $25 \% ?$
e. Answer parts $(\mathrm{a}),(\mathrm{b}),(\mathrm{c}),$ and $(\mathrm{d}),$ using the applet Normal Tail Areas and Quantiles.
f. If it is known that a student's score exceeds $72,$ what is the probability that his or her score exceeds 84?

Key Concepts

Conditional Probability in the Context of Normal Distributions

Conditional probability in a normal distribution involves calculating the probability of an event occurring given that another event has already occurred. This concept is crucial when additional information is available about the outcome, and it adjusts the probability estimation accordingly.

Use of Statistical Software and Applets

Statistical software and interactive applets designed for normal distributions, such as those for computing tail areas and quantiles, provide practical tools for visualizing and calculating probabilities, standard scores, and cutoff values. These tools complement theoretical approaches and are particularly useful for handling complex or conditional probability calculations.

Tail Probabilities

Tail probabilities refer to the areas in the extreme ends (either left or right) of the distribution curve. These probabilities are important for making decisions such as determining the likelihood that a variable exceeds (or falls below) a certain value, which is essential in hypothesis testing and risk assessment.

Quantile Calculation

Quantile calculation involves determining the value below which a certain percentage of the data in a distribution falls. In the context of the normal distribution, quantiles are used to identify threshold scores corresponding to specified cumulative probabilities, which is critical for classification tasks such as grading or setting cutoff points.

Standardization (Z-scores)

Standardization is the process of converting a normally distributed variable into a standard normal variable (or Z-score) by subtracting the mean and dividing by the standard deviation. This transformation simplifies the computation of probabilities and quantiles by allowing the use of standard normal distribution tables or software regardless of the original variable’s scale.

Normal Distribution

This is a continuous probability distribution that is symmetric about its mean and shows that data near the mean are more frequent in occurrence than data far from the mean. The normal distribution is important because many real-world variables are approximately normally distributed, and its properties allow for the use of standard statistical techniques for probability and inference.

Transcript

00:01 So we have a test, an examination that has a normal distribution and it has a mean of 78 and a variance of 36.

00:10 So its standard deviation is 6.

00:12 And we want to know in part a, what is the probability that a person taking the exam scores higher than 72? and that's going to be to have a z -score.

00:23 That is 72 minus 78 divided by 6 is going to be a z -score of negative 1.

00:31 And so that probability of greater than negative 1 is the same as that below positive 1.

00:38 And that is .8413.

00:42 Part b asks, suppose that a student is scoring in the top 10 % receives an a.

00:52 And we want to know what would be the minimum score to get that a.

00:57 Well, this z -score corresponds to a z -value of about 1 .28.

01:05 So if we take the mean of 78 and add on 1 .28 standard deviations and a standard deviation of 6, that will give us that cutoff point.

01:18 So 78 plus 1 .28 times 6 would be a score of about 85 .68 and you can round that how you need.

01:29 Now on part c, it says what must be the cutoff for passing the exam if only the top will be passing? only the top .281 of all scores are going to pass.

01:49 So we need to find what that value is.

01:53 And you can look that value up in your table.

01:55 You can also use an inverse normal, by the way, to find that value...

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