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
asked 2021-03-04
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?

Answers (1)

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.
0

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