(a) The population of a city increased from 23,400 to 27,800 between 2008 and 2012. We need to find the change of population per year, assuming the change was constant from 2008 to 2012. (b) Nelwin's company has a fixed cost of K1,250 per month and a production cost of K37.50 per item. We need to write a linear function $C(x)$ for the cost of producing $x$ items in a month and calculate the monthly cost for producing 100 items. (c) We are given a table of chi-square values for various $n$ and $p$. We need to estimate the chi-square value when $n = 23$ and $p = 99$. (d) We need to find the number of possible car registration plate numbers in PNG, where repetitions of any character are not allowed, and the car registration has the first 3 letters followed by the last 3 digits.

1. Problem Description

(a) The population of a city increased from 23,400 to 27,800 between 2008 and

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2. We need to find the change of population per year, assuming the change was constant from 2008 to

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2. (b) Nelwin's company has a fixed cost of K1,250 per month and a production cost of K37.50 per item. We need to write a linear function $C(x)$ for the cost of producing $x$ items in a month and calculate the monthly cost for producing 100 items.

(c) We are given a table of chi-square values for various $n$ and $p$. We need to estimate the chi-square value when $n = 23$ and $p = 99$.

(d) We need to find the number of possible car registration plate numbers in PNG, where repetitions of any character are not allowed, and the car registration has the first 3 letters followed by the last 3 digits.

2. Solution Steps

(a)

First, calculate the total population increase:

$27,800 - 23,400 = 4,400$.

Next, calculate the number of years between 2008 and 2012:

$2012 - 2008 = 4$ years.

Then, divide the total population increase by the number of years to find the annual change:

$\frac{4,400}{4} = 1,100$.

(b)

The linear function for the cost $C(x)$ of producing $x$ items is given by the sum of the fixed cost and the variable cost:

$C(x) = \text{Fixed Cost} + (\text{Cost per item} \times \text{Number of items})$.

$C(x) = 1250 + 37.50x$.

To calculate the monthly cost for producing 100 items, substitute $x = 100$ into the equation:

$C(100) = 1250 + 37.50(100) = 1250 + 3750 = 5000$.

(c)

We are given a table with values for $n = 20, 22, 24$ and $p = 95, 99, 99.9$.

We want to estimate the chi-square value for $n = 23$ and $p = 99$.

We can use linear interpolation between the values for $n = 22$ and $n = 24$ when $p=99$.

The chi-square value for $n=22, p=99$ is 40.

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9. The chi-square value for $n=24, p=99$ is 42.

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Since 23 is halfway between 22 and 24, we can take the average of the two values:

$\frac{40.29 + 42.98}{2} = \frac{83.27}{2} = 41.635$.

So, the estimated chi-square value for $n = 23$ and $p = 99$ is approximately 41.

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(d)

There are 26 letters in the English alphabet.

Since repetitions are not allowed, for the first letter, there are 26 choices.

For the second letter, there are 25 choices.

For the third letter, there are 24 choices.

There are 10 digits (0-9). Since repetitions are not allowed:

For the first digit, there are 10 choices.

For the second digit, there are 9 choices.

For the third digit, there are 8 choices.

The total number of possible car registration plate numbers is:

$26 \times 25 \times 24 \times 10 \times 9 \times 8 = 11,232,000$.

3. Final Answer

(a) 1,100

(b) $C(x) = 1250 + 37.50x$; K5,000

(c) 41.64

(d) 11,232,000