Use the central limit theorem to find the probability that the sample mean is between two given real numbers Statistics & Probabilities

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(564 \lt m \lt 600\right)

Given that \mu =582, \sigma = 65 and n=25, the corresponding z-scores are:

P\left(\dfrac{564-582}{65/\sqrt{25}} \lt Z \lt \dfrac{600-582}{65/\sqrt{25}}\right) =P\left( -1.38 \lt Z \lt 1.38\right)

This probability can also be rewritten as:

P\left(Z \lt 1.38\right) - P\left(Z\lt-1.38\right) = P\left(Z \leq 1.38\right) - P\left(Z\leq-1.38\right)

P\left(Z \leq 1.38\right) - P\left(Z\leq-1.38\right) =F\left(1.38\right)-F\left(-1.38\right)

By the z-table we have:

F\left(1.38\right)-F\left(-1.38\right) = 0.83

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(358 \lt m \lt 370\right)

Given that \mu =364, \sigma = 35 and n=49, the corresponding z-scores are:

P\left(\dfrac{358-364}{35/\sqrt{49}} \lt Z \lt \dfrac{370-564}{35/\sqrt{49}}\right) =P\left( -1.2 \lt Z \lt 1.2\right)

This probability can also be rewritten as:

P\left(Z \lt 1.2\right) - P\left(Z\lt-1.2\right) = P\left(Z \leq 1.2\right) - P\left(Z\leq-1.2\right)

P\left(Z \leq 1.2\right) - P\left(Z\leq-1.2\right) =F\left(1.2\right)-F\left(-1.2\right)

By the z-table we have:

F\left(1.2\right)-F\left(-1.2\right) = 0.8\ 849-0.1\ 151 \approx 0.77

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(100 \lt m \lt 200\right)

Given that \mu =131, \sigma = 80 and n=16, the corresponding z-scores are:

P\left(\dfrac{100-131}{80/\sqrt{16}} \lt Z \lt \dfrac{200-131}{80/\sqrt{16}}\right) =P\left( -1.55 \lt Z \lt 3.45\right)

This probability can also be rewritten as:

P\left(Z \lt 3.45\right) - P\left(Z\lt-1.55\right) = P\left(Z \leq 3.45\right) - P\left(Z\leq-1.55\right)

P\left(Z \leq 3.45\right) - P\left(Z\leq-1.55\right) =F\left(3.45\right)-F\left(-1.55\right)

By the z-table we have:

F\left(3.45\right)-F\left(-1.55\right) = 0.9\ 997-0.606 \approx 0.94

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(171 \lt m \lt 189\right)

Given that \mu =177, \sigma = 48 and n=16, the corresponding z-scores are:

P\left(\dfrac{171-177}{48/\sqrt{64}} \lt Z \lt \dfrac{189-177}{48/\sqrt{64}}\right) =P\left( -1 \lt Z \lt 2\right)

This probability can also be rewritten as:

P\left(Z \lt 2\right) - P\left(Z\lt-1\right) = P\left(Z \leq 2\right) - P\left(Z\leq-1\right)

P\left(Z \leq 2\right) - P\left(Z\leq-1\right) =F\left(2\right)-F\left(-1\right)

By the z-table we have:

F\left(2\right)-F\left(-1\right) = 0.9\ 772-0.1\ 587 \approx 0.82

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(1\ 200 \lt m \lt 1\ 300\right)

Given that \mu =1\ 216, \sigma = 660 and n=121, the corresponding z-scores are:

P\left(\dfrac{1\ 200-1\ 216}{660/\sqrt{121}} \lt Z \lt \dfrac{1\ 300-1\ 216}{660/\sqrt{121}}\right) =P\left( -0.27 \lt Z \lt 1.4\right)

This probability can also be rewritten as:

P\left(Z \lt 1.4\right) - P\left(Z\lt-0.27\right) = P\left(Z \leq 1.4\right) - P\left(Z\leq-0.27\right)

P\left(Z \leq 1.4\right) - P\left(Z\leq-0.27\right) =F\left(1.4\right)-F\left(-0.27\right)

By the z-table we have:

F\left(1.4\right)-F\left(-0.27\right) = 0.9\ 192-0.3\ 936 \approx 0.53

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(40 \lt m \lt 45\right)

Given that \mu =43, \sigma = 18 and n=81, the corresponding z-scores are:

P\left(\dfrac{40-43}{18/\sqrt{81}} \lt Z \lt \dfrac{45-43}{18/\sqrt{81}}\right) =P\left( -1.5 \lt Z \lt 1\right)

This probability can also be rewritten as:

P\left(Z \lt 1\right) - P\left(Z\lt-1.5\right) = P\left(Z \leq 1\right) - P\left(Z\leq-1.5\right)

P\left(Z \leq 1\right) - P\left(Z\leq-1.5\right) =F\left(1\right)-F\left(-1.5\right)

By the z-table we have:

F\left(1\right)-F\left(-1.5\right) = 0.8\ 413-0.668 \approx 0.77

We have:

P\left(a \lt m \lt b\right) = P\left(\dfrac{a-\mu}{\sigma/\sqrt{n}} \lt Z \lt \dfrac{b-\mu}{\sigma/\sqrt{n}}\right)

We want to compute the following:

P\left(90 \lt m \lt 100\right)

Given that \mu =77, \sigma = 66 and n=36, the corresponding z-scores are:

P\left(\dfrac{90-77}{66/\sqrt{36}} \lt Z \lt \dfrac{100-77}{66/\sqrt{36}}\right) =P\left( 2.09 \lt Z \lt 1.18\right)

This probability can also be rewritten as:

P\left(Z \lt 2.09\right) - P\left(Z\lt1.18\right) = P\left(Z \leq 2.09\right) - P\left(Z\leq 1.18\right)

P\left(Z \leq 2.09\right) - P\left(Z\leq 1.18\right) =F\left(2.09\right)-F\left(1.18\right)

By the z-table we have:

F\left(2.09\right)-F\left(1.18\right) = 0.9\ 817-0.8\ 810 \approx 0.1