Suppose you randomly guess on every question of a 20 question True/False test. Let x = number of questions that you answer correctly. Although values of x are whole numbers, x has an approximately normal distribution with mean = 10 and standard deviation = 2.2. What is the probability that you would get at least 12 questions correct?

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makenzietayler232 makenzietayler232

13-02-2020
Mathematics

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Suppose you randomly guess on every question of a 20 question True/False test. Let x = number of questions that you answer correctly. Although values of x are whole numbers, x has an approximately normal distribution with mean = 10 and standard deviation = 2.2. What is the probability that you would get at least 12 questions correct?

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joaobezerra joaobezerra · Answer 1 · 2020-02-13T22:27:08+01:00

Answer:

18.14% probability that you would get at least 12 questions correct.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 10, \sigma = 2.2[/tex]

What is the probability that you would get at least 12 questions correct?

This is 1 subtracted by the pvalue of Z when X = 12. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{12 - 10}{2.2}[/tex]

[tex]Z = 0.91[/tex]

[tex]Z = 0.91[/tex] has a pvalue of 0.8186.

So there is a 1-0.8186 = 0.1814 = 18.14% probability that you would get at least 12 questions correct.