Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True. In the table, n is the size of the sample and $ar{p}$ is the sample proportion. | Statement | Always True | Sometimes True | Never True | |-----------|-------------|----------------|------------| | 1. When forming confidence intervals (for fixed n and $ar{p}$), an increased confidence level implies a wider interval. ||| | | 2. As the value of $ar{p}$ increases (for fixed n), the estimated standard error of the population proportion increases. ||| | | 3. As the value of $ar{p}(1 - ar{p})$ increases (for fixed n), the estimated standard error of the population proportion increases. ||| | | 4. As n, the number of people sampled, increases (for fixed $ar{p}$), the estimated standard error of the population proportion increases. ||| | (ii) Using calculus or otherwise, find the maximum value of $ar{p}(1 - ar{p})$. (iii) Hence, find the largest possible value of the radius of the 95% confidence interval for a population proportion, given a random sample of size 800.

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Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True - Leaving Cert Mathematics - Question b - 2018

Question b

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Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True. In the table, n is the size of the sam... show full transcript

Worked Solution & Example Answer:Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True - Leaving Cert Mathematics - Question b - 2018

Step 1

1. When forming confidence intervals (for fixed n and $ar{p}$), an increased confidence level implies a wider interval.

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Answer

This statement is Always True. A higher confidence level leads to a wider interval as it requires capturing a larger amount of variability in the population.

Step 2

2. As the value of $ar{p}$ increases (for fixed n), the estimated standard error of the population proportion increases.

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Answer

This statement is Sometimes True. The standard error is calculated as $SE = rac{ar{p}(1 - ar{p})}{ ext{sqrt}(n)}$ which means the standard error is at a maximum when $ar{p} = 0.5$ ; thus, it does not necessarily increase with $ar{p}$ .

Step 3

3. As the value of $ar{p}(1 - ar{p})$ increases (for fixed n), the estimated standard error of the population proportion increases.

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Answer

This statement is Always True. The term $ar{p}(1 - ar{p})$ directly affects the standard error, thus as it increases, so does the error.

Step 4

4. As n, the number of people sampled, increases (for fixed $ar{p}$), the estimated standard error of the population proportion increases.

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Answer

This statement is Never True. As n increases, the standard error decreases because it is inversely proportional to the square root of n.

Step 5

(ii) Using calculus or otherwise, find the maximum value of $ar{p}(1 - ar{p})$.

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Answer

To find the maximum value of $ar{p}(1 - ar{p})$ , we can consider the function:

$f(ar{p}) = ar{p} - ar{p}^2$

Taking the derivative:

$f'(ar{p}) = 1 - 2ar{p}$ Setting the derivative equal to zero gives:

ightarrow ar{p} = rac{1}{2}$$ Plugging back to find the maximum: $$M = ar{p}(1 - ar{p}) = rac{1}{2}(1 - rac{1}{2}) = rac{1}{4} = 0.25$$

Step 6

(iii) Hence, find the largest possible value of the radius of the 95% confidence interval for a population proportion, given a random sample of size 800.

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Answer

Considering the maximum of eigenvalue $M$ processed earlier, the standard error is calculated as:

$SE = rac{ar{p}(1 - ar{p})}{ ext{sqrt}(n)}$

With $ar{p} = rac{1}{2}$ and $n = 800$ :

= rac{0.25}{28.2843} \ ightarrow SE ext{ approximately equals } 0.0088$$ For a 95% confidence interval, we use: $$z = 1.96$$ Thus, the radius of the confidence interval is: $$R = z imes SE = 1.96 imes 0.0088 \ ightarrow R ext{ approximately equals } 0.0173 ext{ or } 1.73 ext{%.}$$

Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True - Leaving Cert Mathematics - Question b - 2018

Worked Solution & Example Answer:Put one tick into the table for each statement to indicate whether the statement is Always True, Sometimes True or Never True - Leaving Cert Mathematics - Question b - 2018

1. When forming confidence intervals (for fixed n and $ar{p}$), an increased confidence level implies a wider interval.

2. As the value of $ar{p}$ increases (for fixed n), the estimated standard error of the population proportion increases.

3. As the value of $ar{p}(1 - ar{p})$ increases (for fixed n), the estimated standard error of the population proportion increases.

4. As n, the number of people sampled, increases (for fixed $ar{p}$), the estimated standard error of the population proportion increases.

(ii) Using calculus or otherwise, find the maximum value of $ar{p}(1 - ar{p})$.

(iii) Hence, find the largest possible value of the radius of the 95% confidence interval for a population proportion, given a random sample of size 800.

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1. When forming confidence intervals (for fixed n and $ar{p}$), an increased confidence level implies a wider interval.

2. As the value of $ar{p}$ increases (for fixed n), the estimated standard error of the population proportion increases.

3. As the value of $ar{p}(1 - ar{p})$ increases (for fixed n), the estimated standard error of the population proportion increases.

4. As n, the number of people sampled, increases (for fixed $ar{p}$), the estimated standard error of the population proportion increases.

(ii) Using calculus or otherwise, find the maximum value of $ar{p}(1 - ar{p})$.