Normal Probability Distribution

$X$ has Normal distribution with mean $2$ and standard deviation $3$. Find the value of the variable $x$ such that the probability of the interval from the mean to that value is $0.4115$.

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.

Sketch the event $P\left(29\le T_L<32\right)$if the length of a local bus journey from ranjith's home to school is a normally distributed random variable $T_L$ with mean $31$ minutes and standard deviation $5$ minutes.

Sketch the event $P\left(T_L\ge 31\right)$ if the length of a local bus journey from Raj' s home to school is a normally distributed random variable $T_L$ with mean $31$ minutes and standard deviation $5$ minutes.

A factory produces half-litre tins of oil. The volume of oil in a tin is normally distributed with mean $506.18\mathrm{ml}$ and standard deviation $2.96\mathrm{ml}$. What percentage of the tins contain less than half a litre of oil? Use $\left[\varphi \left(2.088\right)=0.9816\right]$.

The mass of a newborn baby in a certain region is normally distributed with mean $3.35 \mathrm{kg}$ and variance $0.0858{ \mathrm{kg}}^2$. Estimate how many of the$1356$ babies born last year had masses of less than $3.5 \mathrm{kg}$. Use $\left[\varphi \left(0.512\right)=0.6957\right]$

For $Z~\mathrm{N}(0,1),$ find the probability $P (Z\ge -0.077)$

For $Z~\mathrm{N}(0,1),$ find the probability of $P (Z>-1.53)$

For $Z~\mathrm{N}(0,1),$ find the probability of $P (Z\le 2.468)$. Use $\varphi \left(2.46\right)\approx 0.993$

For $Z~\mathrm{N}(0,1),$ find the probability of $P (Z<0.567)$

The values in two datasets, whose probability distributions are both normal curves, are summarised by the following totals: On the same diagram, sketch a normal curve for each dataset.
$\Sigma x^2=35000, \Sigma x=12000$ and $n=5000.$
$\Sigma y^2=72000, \Sigma y=26000$ and $n=10000.$

$X~Po\left(42\right)$. Use the normal approximation to find: $P\left(X<50\right)$

State the null and alternative hypotheses and test statistic for a test for the population mean for the random variable $X~N\left(42, 8\right)$, sample value $45$; two-tailed test at $10\%$ level of significance.

Giving your answers to $3$ significant figures, evaluate:

$e^{-9}$

Giving your answers to $3$ significant figures, evaluate:

$e^{-3.6}$

Giving your answers to $3$ significant figures, evaluate:

$e^{-2}$

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)

Given that $X~B\left(120, 0.55\right)$, state a suitable approximating distribution and use it to find:

$P\left(X⩽63\right)$