The Normal Distribution

Find the probability that the mass of pasta in a randomly chosen large bag is less than 2.65 kg.

A restaurant manager buys 160 of these large bags of pasta.

Find the number of bags for which you would expect the mass of pasta to be more than 1.65 standard deviations above the mean.

The masses of small bags of pasta sold by the company are normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. Tests show that 77% of these bags have masses greater than 1.26 kg, and 44% have masses less than 1.35 kg.

Find, in either order, the value of $\mu$ and the value of $\sigma$.

For 150 randomly chosen trees from this forest, how many would you expect to be classified as small?

Trees from this forest are classified as tall if their height is at least $h\text{m}$. 25% of the trees are classified as tall.

Find the value of $h$.

A factory produces chocolate bars. The weights of the bars are normally distributed with mean $155\text{ g}$ and standard deviation $6\text{ g}$. A random sample of 350 of these bars is chosen.

How many of these 350 bars would you expect to weigh between $148\text{ g}$ and $160\text{ g}$?

A second factory also produces chocolate bars. The weights of these bars are normally distributed with mean $\mu\text{ g}$ and standard deviation $\sigma\text{ g}$. Tests show that $8\%$ of the bars weigh more than $114.0\text{ g}$ and $20\%$ weigh less than $99.5\text{ g}$.

Find the value of $\mu$ and the value of $\sigma$.

How many members of the club would you expect to have heights within 5 cm of the mean?

The heights of the Junior members of the Giraffes club are normally distributed with mean 172.7 cm and standard deviation $\sigma$ cm. 23% of these members have height less than 170.3 cm.

Find the value of $\sigma$.

Find the probability that, on a randomly chosen day, Obi takes more than 15.6 minutes to walk to work.

On 90% of days, Obi takes more than $t$ minutes to walk to work.

Find the value of $t$.

Obi walks to work 5 days a week for 45 weeks in a year.

On how many days in a year would you expect Obi to take within one minute of the mean time to walk to work?

Find the probability that a randomly chosen runner took more than 55.1 seconds to complete the run.

In the second run, the times to complete the run were normally distributed with mean 39.8 seconds and standard deviation $\sigma$ seconds. 10% of the runners took more than 48.6 seconds.

Find the value of $\sigma$.

150 runners are chosen at random from those who took part in the second run.

Use an approximation to find the probability that fewer than 20 of the 150 runners took more than 48.6 seconds to complete the run.

In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than $1.26\text{ kg}$?

The masses of the large bags of rice are normally distributed with mean $2.50\text{ kg}$ and standard deviation $\sigma\text{ kg}$. 20% of these large bags of rice have a mass less than $2.40\text{ kg}$.

Find the value of $\sigma$.

A random sample of 80 large bags of rice is chosen.

Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than $2.40\text{ kg}$.

Find the probability that a randomly chosen adult raccoon of this species has a tail length between $23\text{ cm}$ and $35\text{ cm}$.

The masses of adult raccoons of this species are normally distributed with mean $8.5\text{ kg}$ and standard deviation $\sigma\text{ kg}$. 75% of adult raccoons of this species have mass greater than $7.6\text{ kg}$.

Find the value of $\sigma$.