The Distribution of Sample Means

Find the probability that for $16$ randomly selected calls made to the centre, the mean time taken to answer the calls is less than $18$ seconds. The time taken for telephone calls to a call centre to be answered is normally distributed with mean $20$ seconds and standard deviation $5$ seconds.

The random variable $\overline{Y}$ is the mean of a random sample of $50$ observations of $Y$. The random variable $Y$ has mean $21$ and standard deviation $4.2$. State the approximate distribution of $\overline{Y}$, giving its parameters, and work out $P\left(\overline{Y}<22\right)$.

The random variable $\overline{X}$ is the mean of a random sample of $100$ observations of $X$. The random variable $X$ has mean $30$ and variance $36$. State the approximate distribution of $\overline{X}$, giving its parameters, and find the probability that the sample mean is greater than $31$.

The random variable $\overline{X}$ is the mean of a random sample of $80$ observations of $X$. The random variable $X$ has mean $6$ and variance $8$. State the approximate distribution of $\overline{X}$, giving its parameters, and find the probability that the sample mean is less than $6.4$.

A random sample of size $60$ is taken from the random variable $X$, where $X~B\left(45, 0.4\right)$. Given that $\overline{X}$ is the sample mean, find:

$P\left(\overline{X}<19\right)$

(Correct the answer up to two decimal place)

The lengths of time people take to complete a certain type of puzzle are normally distributed with mean $48.8$ minutes and standard deviation $15.6$ minutes. The random variable $X$ represents the time taken, in minutes, by a randomly chosen person to solve this type of puzzle. The times taken by random samples of $5$ people are noted. The mean time $\overline{X}$ is calculated for each sample.

Find $P\left(\overline{X}<50\right)$.(Upto three decimal places)

It is known that the number, $N$, of words contained in the leading article each day in a certain newspaper can be modelled by a normal distribution with mean $352$ and variance $29$. A researcher takes a random sample of $10$ leading articles and finds the sample mean, $\overline{N}$, of $N$.

Find $P\left(\overline{N}>354\right)$.

(Correct the answer up to three decimal place)

A random sample of $35$ observations is to be taken from a normal distribution with mean $15$ and variance $9$. If $\overline{X}$ is the sample mean, find:

the value of $k$, where $P\left(\overline{X}<k\right)=0.75$.

A random sample of $35$ observations is to be taken from a normal distribution with mean $15$ and variance $9$. If $\overline{X}$ is the sample mean, find $P\left(\overline{X}<16.2\right)$.

(Correct the answer up to three decimal place)

The burn time, in minutes, for a certain brand of candle can be modelled by a normal distribution with mean $90$ and standard deviation $15.6$. Find the probability that a random sample of five candles, each one lit immediately after another burns out, will burn for a total of $500$ minutes or less.

(Correct the answer up to three decimal place)

The score on a four-sided spinner is given by the random variable $X$ with probability distribution as shown in the table.

Show that the variance is $1.01$.

The mean and standard deviation of the time spent by visitors at an art gallery are $3.5$ hours and $1.5$ hours, respectively.

Find the probability that the mean time spent in the art gallery by a random sample of: $20$ people is less than $4$ hours. (Correct the answer up to three decimal place)

The mean and standard deviation of the time spent by visitors at an art gallery are $3.5$ hours and $1.5$ hours, respectively.

Find the probability that the mean time spent in the art gallery by a random sample of $60$ people is more than $4$ hours.

(Correct the answer up to four decimal place)

A random sample of size $50$ is taken from random variable $X$, where $X~\mathit{Po}\left(2\right)$. Find $P\left(1.5<\overline{X}⩽2.2\right)$, where $\overline{X}$ is the sample mean. (Correct the answer up to three decimal place)

A random sample of size $60$ is taken from the random variable $X$, where $X~B\left(45, 0.4\right)$. Given that $\overline{X}$ is the sample mean, find: $P\left(\overline{X}⩽18\right)$  (Correct the answer up to three decimal place)

Ciara needs $5 \mathrm{kg}$ of flour, so she buys $10$ bags, each labelled as containing $500 \mathrm{g}$. Unknown to her, the bags contain, on average, $510 \mathrm{g}$ with variance $120{ \mathrm{g}}^2$. What is the probability that Ciara actually buys less flour than she needs?

(Correct the answer up to three decimal place)

The time taken for telephone calls to a call centre to be answered is normally distributed with mean $20$ seconds and standard deviation $5$ seconds. Find the probability that for $16$ randomly selected calls made to the centre, the mean time taken to answer the calls is less than $18$ seconds.  (Correct your answer up to four decimal places).

The random variable $Y$ has mean $21$ and standard deviation $4.2$. The random variable $\overline{Y}$ is the mean of a random sample of $50$ observations of $Y$. State the approximate distribution of $\overline{Y}$, giving its parameters, and work out $P\left(\overline{Y}<22\right)$.

(Correct the answer up to three decimal place)

The random variable $X$ has mean $30$ and variance $36$. The random variable $\overline{X}$ is the mean of a random sample of $100$ observations of $X$.  find the probability that the sample mean is greater than $31$. (correct the answer up to three decimal places).

The lengths of time people take to complete a certain type of puzzle are normally distributed with mean $48.8$ minutes and standard deviation $15.6$ minutes. The random variable $X$ represents the time taken, in minutes, by a randomly chosen person to solve this type of puzzle. The times taken by random samples of $5$ people are noted. The mean time $\overline{X}$ is calculated for each sample.

State the distribution of $\overline{X}$, giving the values of any parameters.