**Normal Distribution**

Practice Problems

**Question no **

A factory makes metal bars. Their lengths are assumed to be normally distributed with a mean of 180 cm and a standard deviation of 5 cm.

On the following diagram, shade the region representing the probability that a metal bar, chosen at random, will have a length less than 175 cm.

(i) The probability that the length of the metal bar is less than 175 cm is equal to the probability that the length is greater than h cm. Write down the value of h.

(ii) Find the probability that the length of the metal bar is greater than one standard deviation above the mean.

**1. Logic**

Question no: 1

Applicants for a job had to complete a mathematics test. The time they took to complete the test is normally distributed with a mean of 53 minutes and a standard deviation of 16.3. One of the applicants is chosen at random.

For 11% of the applicants it took longer than k

$k$minutes to complete the test.

There were 400 applicants for the job.

Find the probability that this applicant took at least 40 minutes to complete the test.

Find the value of k

$k$.

Estimate the number of applicants who completed the test in less than 25 minutes.

## Markscheme

0.787 (0.787433…, 78.7%) *(M1)(A1) (C2)*

**Note: **Award ** (M1) **for a correct probability statement, P(X>40)

, or a correctly shaded normal distribution graph.

*[2 marks]*

73.0 (minutes) (72.9924…) *(M1)(A1) (C2)*

**Note: **Award ** (M1) **for a correct probability statement, P(X>k)=0.11

, or a correctly shaded normal distribution graph.

*[2 marks]*

0.0423433…×400

$0.0423433\dots \times 400$ *(M1)*

**Note: **Award ** (M1) **for multiplying a probability by 400. Do not award

**for 0.11×400**

*(M1)*.

Use of a lower bound less than zero gives a probability of 0.0429172….

=16

$=16$ *(A1) (C2)*

**Notes: **Accept a final answer of 17. Do not accept a final answer of 18. Accept a non-integer final answer either 16.9 (16.9373…) from use of lower bound zero or 17.2 (17.1669…) from use of the default lower bound of −1099

.

*[2 marks]*

**Question no : 2**

The mass of a certain type of Chilean corncob follows a normal distribution with a mean of 400 grams and a standard deviation of 50 grams.

A farmer labels one of these corncobs as premium if its mass is greater than a

$a$grams. 25% of these corncobs are labelled as premium.

Write down the probability that the mass of one of these corncobs is greater than 400 grams.

Find the value of a

$a$.

Estimate the interquartile range of the distribution.

**Question no : 4**

Date | May 2016 | Marks available | 1 | Reference code | 16M.1.sl.TZ1.8 |

Level | SL only | Paper | 1 | Time zone | TZ1 |

Command term | Write down | Question number | 8 | Adapted from | N/A |

## Question

The lifetime, L

$L$ , of light bulbs made by a company follows a normal distribution.

L

is measured in hours. The normal distribution curve of L

$L$is shown below.

Write down the mean lifetime of the light bulbs.

The standard deviation of the lifetime of the light bulbs is 850

$850$hours.

Find the probability that 5000⩽L⩽6000

$5000\u2a7dL\u2a7d6000$, for a randomly chosen light bulb.

The company states that 90%

$90\mathrm{\%}$of the light bulbs have a lifetime of at least k

$k$hours.

Find the value of k

$k$. Give your answer correct to the nearest hundred.

## Markscheme

**Question no : 2**

**Question no : 2**

**Question no : 2**

**Question no : 2**

**Question no : 2**

Solve the equation \({\log _2}(x + 3) + {\log _2}(x – 3)=4\)