Calculate the sample proportion (p).

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1. Determine a 98% confidence interval for the proportion for PDP in the population.

Calculate the sample proportion ( $\hat{p}$ ).
The number of voters interviewed is $n = 1000$ .
The number of voters who support PDP is $x = 349$ .
$\hat{p} = \frac{x}{n} = \frac{349}{1000} = 0.349$
Find the critical z-value ( $z_{\alpha/2}$ ).
For a 98% confidence interval, $\alpha = 1 - 0.98 = 0.02$ . So, $\alpha/2 = 0.01$ .
The critical z-value corresponding to a cumulative probability of $1 - 0.01 = 0.99$ is $z_{0.01} \approx 2.326$ .
Calculate the standard error ( $SE$ ).
$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.349(1-0.349)}{1000}} = \sqrt{\frac{0.349 \times 0.651}{1000}} = \sqrt{\frac{0.227199}{1000}} = \sqrt{0.000227199} \approx 0.01507$
Calculate the margin of error ( $ME$ ).
$ME = z_{\alpha/2} \times SE = 2.326 \times 0.01507 \approx 0.03505$
Construct the confidence interval.
The confidence interval is $\hat{p} \pm ME$ .
$0.349 \pm 0.03505$ $(0.349 - 0.03505, 0.349 + 0.03505)$ $(0.31395, 0.38405)$
Rounding to three decimal places:
$(0.314, 0.384)$
The 98% confidence interval for the proportion of PDP supporters is $\boxed{\mathbf{(0.314, 0.384)}}$ .

2. Construct a 95% confidence interval for the proportion $P$ of defective items produced by the factory.
Calculate the sample proportion ( $\hat{p}$ ).
The number of products sampled is $n = 75$ .
The number of defective products is $x = 11$ .
$\hat{p} = \frac{x}{n} = \frac{11}{75} \approx 0.146667$
Find the critical z-value ( $z_{\alpha/2}$ ).
For a 95% confidence interval, $\alpha = 1 - 0.95 = 0.05$ . So, $\alpha/2 = 0.025$ .
The critical z-value corresponding to a cumulative probability of $1 - 0.025 = 0.975$ is $z_{0.025} \approx 1.96$ .
Calculate the standard error ( $SE$ ).
$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.146667(1-0.146667)}{75}} = \sqrt{\frac{0.146667 \times 0.853333}{75}} = \sqrt{\frac{0.125111}{75}} = \sqrt{0.00166814} \approx 0.04084$
Calculate the margin of error ( $ME$ ).
$ME = z_{\alpha/2} \times SE = 1.96 \times 0.04084 \approx 0.08005$
Construct the confidence interval.
The confidence interval is $\hat{p} \pm ME$ .
$0.146667 \pm 0.08005$ $(0.146667 - 0.08005, 0.146667 + 0.08005)$ $(0.066617, 0.226717)$
Rounding to three decimal places:
$(0.067, 0.227)$
The 95% confidence interval for the proportion of defective items is $\boxed{\mathbf{(0.067, 0.227)}}$ .

3. Compute a 96% confidence interval for the proportion of people who use Knorr Maggi cube.
Calculate the sample proportion ( $\hat{p}$ ).
The number of people questioned is $n = 320$ .
The number of people who answered 'yes' is $x = 82$ .
$\hat{p} = \frac{x}{n} = \frac{82}{320} = 0.25625$
Find the critical z-value ( $z_{\alpha/2}$ ).
For a 96% confidence interval, $\alpha = 1 - 0.96 = 0.04$ . So, $\alpha/2 = 0.02$ .
The critical z-value corresponding to a cumulative probability of $1 - 0.02 = 0.98$ is $z_{0.02} \approx 2.054$ .
Calculate the standard error ( $SE$ ).
$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.25625(1-0.25625)}{320}} = \sqrt{\frac{0.25625 \times 0.74375}{320}} = \sqrt{\frac{0.1905078125}{320}} = \sqrt{0.000595337} \approx 0.02440$
Calculate the margin of error ( $ME$ ).
$ME = z_{\alpha/2} \times SE = 2.054 \times 0.02440 \approx 0.05011$
Construct the confidence interval.
The confidence interval is $\hat{p} \pm ME$ .
$0.25625 \pm 0.05011$ $(0.25625 - 0.05011, 0.25625 + 0.05011)$ $(0.20614, 0.30636)$
Rounding to three decimal places:
$(0.206, 0.306)$
The 96% confidence interval for the proportion of people who use Knorr Maggi cube is $\boxed{\mathbf{(0.206, 0.306)}}$ .

Calculate the sample proportion (p).

Step-by-step solution

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).

Construct the confidence interval.

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).

Construct the confidence interval.

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).

Construct the confidence interval.

Still stuck on this one?

More Mathematics Questions

Calculate the sample proportion (p).

Step-by-step solution

Calculate the sample proportion (p^\hat{p}p^​).

Find the critical z-value (zα/2z_{\alpha/2}zα/2​).

Calculate the standard error (SESESE).

Calculate the margin of error (MEMEME).

Construct the confidence interval.

Calculate the sample proportion (p^\hat{p}p^​).

Find the critical z-value (zα/2z_{\alpha/2}zα/2​).

Calculate the standard error (SESESE).

Calculate the margin of error (MEMEME).

Construct the confidence interval.

Calculate the sample proportion (p^\hat{p}p^​).

Find the critical z-value (zα/2z_{\alpha/2}zα/2​).

Calculate the standard error (SESESE).

Calculate the margin of error (MEMEME).

Construct the confidence interval.

Still stuck on this one?

More Mathematics Questions

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).

Calculate the sample proportion ( $\hat{p}$ ).

Find the critical z-value ( $z_{\alpha/2}$ ).

Calculate the standard error ( $SE$ ).

Calculate the margin of error ( $ME$ ).