Normal Distribution

If the probability that the length of a randomly chosen chord of a circle lies between $\frac{1}{2}$ and $\frac{3}{4}$ of its diameter is $\frac{m}{n}.$ Where $G.C.D.\left\{m, n\right\}=1.$ Then which of the following is/are true?

In a normal distribution, $31\%$ of the items are under $45$ and $8\%$ are over $64$. Determine the mean and variance of the distribution.

In a normal distribution, $7\%$ of the items are under $35$ and $89\%$ are under $63$. Determine the variance of the distribution.

A circle is inscribed in a square. Point $Q$ in the square is chosen at random. What is the probability that $Q$ lies in the shaded region?

[Take $\mathrm{\pi }=\frac{22}{7}$]

$ABCD$ is a square. $M$ is the midpoint of $BC$ and $N$ is the midpoint of $CD$. A point is selected at random in the square. Calculate the probability that it lies in the triangle $MCN$.

A national park houses a large number of tall trees. The heights of the trees are normally distributed with mean of $45$ metres and a standard deviation of $9$ metres. One tree is selected at random from the park. Given that this tree is at least $55$ metres, find the probability that it is taller than $65$ metres.

Use the standard values:

$$\begin{gathered} P\left(0<Z\le 2.22\right)=0.4868 \\ P\left(0<Z\le 1.11\right)=0.3665 \end{gathered}$$

Find the mean and standard deviation in which $8\%$ items are under the $72$ and $99\%$ are over for the $53$.

A normal distribution has a mean measure $150$ and standard deviation as $10$. Find the value of middle $60\%$ of limits lies between which numbers.

Suppose $X$ is normal with mean $10$. If $P(X>16)=0.0228$, then what is the variance of $X$? [Use $\Phi \left(2\right)=0.9772$]

Suppose $X$ is normal with mean $8$. If $P(X>16)=0.0228$, then what is the variance of $X$? [Use $\Phi \left(2\right)=0.9772$]

Suppose $X$ is normal with mean $4$. If $P(X>16)=0.0228$, then what is the variance of $X$? [Use $\Phi \left(2\right)=0.9772$]

Suppose $X$ is normal with mean $2$. If $P(X>16)=0.0228$, then what is the variance of $X$? [Use $\Phi \left(2\right)=0.9772$]

Suppose $X$ is normal with mean $6$. If $P(X>16)=0.0228$, then what is the variance of $X$? [Use $\Phi \left(2\right)=0.9772$]

If $X~N\left(16, 1.8\right)$, find: $P\left(13<X<15\right)$ upto four decimal places.

If $X~N\left(16, 1.8\right)$, find: $P\left(X<20\right)$

(Correct the answer up to four decimal place)

The masses of cans of baked beans are normally distributed, with a mean mass of $415 \mathrm{g}$ and standard deviation $12 \mathrm{g}$. Ashok buys $144$ cans of beans from the supermarket. What is the probability that at least $75$ of them have a mass of less than $413.5 \mathrm{g}$ ? (Use $P\left(Z<-0.125\right)=0.450$).

The masses of cans of baked beans are normally distributed, with a mean mass of $415 \mathrm{g}$ and standard deviation $12 \mathrm{g}$. A can of beans is randomly selected. It is known that the mass of the can is more than $420 \mathrm{g}$. What is the probability that it weighs more than $422.5 \mathrm{g}$? (Use $P\left(Z<0.625\right)=0.735, P\left(Z<0.417\right)=0.663$).

The masses of cans of baked beans are normally distributed, with a mean mass of $415 \mathrm{g}$ and standard deviation $12 \mathrm{g}$. The probability that a randomly selected can of beans has a mass greater than $m \mathrm{g} \text{is} 0.65$. Find the value of $m$

The times taken by Blossom to walk from her home to her local cafe are normally distributed, with a mean time of $35$ minutes and a standard deviation of $3.4$ minutes. Blossom walks to her cafe on $25$ separate occasions. On how many occasions should she expect the journey to take less than $30$ minutes? (Use $P\left(Z<-1.47\right)=0.0708$).

The times taken by Blossom to walk from her home to her local cafe are normally distributed, with a mean time of $35$ minutes and a standard deviation of $3.4$ minutes. Find the probability that on a randomly selected day, Blossom's journey takes between $34$ and $36.5$ minutes (Use $P\left(Z<-0.29\right)=0.38591, P\left(Z<0.44\right)=0.67003$).