# Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?

Question
Normal distributions
Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?

2021-03-05
Step 1
Suppose, X be the marks of statistics students, X follows Normal distribution with mean 72 and standard deviation 9.
The average marks of 10 students,
$$\displaystyle\overline{{X}}=\frac{{1}}{{10}}{\sum_{{{i}={1}}}^{{10}}}{X}_{{i}}$$,
$$\displaystyle\overline{{X}}$$ follows Normal distributions with mean 72 and standard deviation $$\displaystyle\frac{{9}}{{\sqrt{{10}}}}$$.
Step 2
The probability that the sample mean between 71 and 73 is
$$\displaystyle{P}{\left({71}\le\overline{{X}}\le{73}\right)}$$
$$\displaystyle={P}{\left(\frac{{{71}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\le{\left(\frac{{\overline{{X}}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\le{\left(\frac{{{73}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\right)},{Z}={\left(\frac{{\overline{{X}}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\sim{N}{\quad\text{or}\quad}{m}{a}{l}{\left({0},{1}\right)}\right.}\right.}\right.}$$
$$\displaystyle={P}{\left(-{0.3514}\le{Z}\le{0.3514}\right)}$$
$$\displaystyle={P}{\left({Z}\le{0.3514}\right)}-{P}{\left({Z}\le-{0.3514}\right)}$$
$$\displaystyle=\Phi{\left({0.3514}\right)}-\Phi{\left({0.3514}\right)},\Phi{\left({z}\right)}={P}{\left({Z}\le{z}\right)}$$
$$\displaystyle={2}\Phi{\left({0.3514}\right)}-{1},\Phi{\left({z}\right)}={1}-\Phi{\left(-{z}\right)}$$
$$\displaystyle={\left({2}\times{0.6374}\right)}-{1}$$
=0.2748
The values of $$\displaystyle\Phi{\left({0.3514}\right)}$$ is taken from Normal distribution.
Therefore, the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is 0.2758.

### Relevant Questions

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a.
To compute the confidence interval use a ? distribution.
b.
With $$90\%$$ confidence the population mean minutes of concentration is between ____ and ____ minutes.
c.
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True or False
1.The goal of descriptive statistics is to simplify, summarize, and organize data.
2.A summary value, usually numerical, that describes a sample is called a parameter.
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8.In a distribution with a mean of M = 76 and a standard deviation of SD = 7, a score of 91 would be considered an extreme value.
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