Mathematics, 23.07.2021 05:00 Martinez6354

n a history class there are 88 history majors and 88 non-history majors. 44 students are randomly selected to present a topic. What is the probability that at least 22 of the 44 students selected are non-history majors

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Answer from: tiwaribianca475

0.5675 = 56.75% probability that at least 22 of the 44 students selected are non-history majors.

Step-by-step explanation:

The students are chosen without replacement from the sample, which means that the hypergeometric distribution is used to solve this question. We are working also with a sample with more than 10 history majors and 10 non-history majors, which mean that the normal approximation can be used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 p-value, we get the probability that the value of the measure is greater than X.

Approximation:

We have to use the mean and the standard deviation of the hypergeometric distribution, that is:

$\mu = \frac{nk}{N}$

$\sigma = \sqrt{\frac{nk(N-k)(N-n)}{N^2(N-1)}}$

In this question:

88 + 88 = 176 students, which means that N = 176

88 non-history majors, which means that k = 88

44 students are selected, which means that n = 44

Mean and standard deviation:

$\mu = \frac{44*88}{176} = 22$

$\sigma = \sqrt{\frac{44*88*(176-88)*(176-44)}{176^2(175-1)}} = 2.88$

What is the probability that at least 22 of the 44 students selected are non-history majors?

Using continuity correction, as the hypergeometric distribution is discrete and the normal is continuous, this is $P(X \geq 22 - 0.5) = P(X \geq 21.5)$ , which is 1 subtracted by the p-value of Z when X = 21.5. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{21.5 - 22}{2.88}$

Z = -0.17

Z = -0.17 has a p-value of 0.4325

1 - 0.4325 = 0.5675

0.5675 = 56.75% probability that at least 22 of the 44 students selected are non-history majors.

Answer from: Quest

Where are the functions?

Answer from: Quest

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.

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