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AWFHayami
10.03.2021 •
Mathematics
37% of Americans say that math is the most important subject in school. In a random sample of 400 Americans, what is the probability that between 40% and 45% will say that math is the most important subject?
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Ответ:
0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
and standard deviation ![s = \sqrt{\frac{p(1-p)}{n}}](/tpl/images/1185/1245/73804.png)
37% of Americans say that math is the most important subject in school.
This means that![p = 0.37](/tpl/images/1185/1245/61260.png)
Sample of 400 Americans
This means that![n = 400](/tpl/images/1185/1245/efab9.png)
Mean and standard deviation:
What is the probability that between 40% and 45% will say that math is the most important subject?
This is the pvalue of Z when X = 0.45 subtracted by the pvalue of Z when X = 0.4. So
X = 0.45
By the Central Limit Theorem
X = 0.4
0.9995 - 0.8925 = 0.1070
0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject
Ответ: