Math / Data & Statistics

QuestionSuppose that a simple random sample of size $n=315$ selected from a population has $x=185$ successes. Calculate the margin of error for a $95 \%$ confidence interval for the proportion of successes for the population, $p$ . Compute the sample proportion, $\hat{p}$ , standard error estimate, SE, critical value, $z$ , and the margin of error, $m$ . Use a $z$ -distribution table to determine the critical value. Give all of your answers to three decimal places except give the critical value, $z$ , to two decimal places. $\hat{p}=$ $\square$ $\mathrm{SE}=$ $\square$ $z=$ $\square$ $m=$ $\square$

Studdy Solution

STEP 1

1. The sample size $n = 315$ .
2. The number of successes $x = 185$ .
3. We are using a $95\%$ confidence interval.
4. The population is large enough for the Central Limit Theorem to apply.

STEP 2

1. Calculate the sample proportion $\hat{p}$ .
2. Calculate the standard error estimate $\mathrm{SE}$ .
3. Determine the critical value $z$ for a $95\%$ confidence interval.
4. Calculate the margin of error $m$ .

STEP 3

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{185}{315}$
$\hat{p} \approx 0.587$

STEP 4

Calculate the standard error estimate $\mathrm{SE}$ :
$\mathrm{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.587 \times (1 - 0.587)}{315}}$
$\mathrm{SE} \approx 0.028$

STEP 5

Determine the critical value $z$ for a $95\%$ confidence interval. Using a $z$ -distribution table, the critical value is:
$z \approx 1.96$

STEP 6

Calculate the margin of error $m$ :
$m = z \times \mathrm{SE} = 1.96 \times 0.028$
$m \approx 0.055$
The sample proportion is:
$\hat{p} = 0.587$
The standard error estimate is:
$\mathrm{SE} = 0.028$
The critical value is:
$z = 1.96$
The margin of error is:
$m = 0.055$

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Studdy Solution

STEP 1

1. The sample size $n = 315$ .
2. The number of successes $x = 185$ .
3. We are using a $95\%$ confidence interval.
4. The population is large enough for the Central Limit Theorem to apply.

STEP 2

1. Calculate the sample proportion $\hat{p}$ .
2. Calculate the standard error estimate $\mathrm{SE}$ .
3. Determine the critical value $z$ for a $95\%$ confidence interval.
4. Calculate the margin of error $m$ .

STEP 3

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{185}{315}$
$\hat{p} \approx 0.587$

STEP 4

Calculate the standard error estimate $\mathrm{SE}$ :
$\mathrm{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.587 \times (1 - 0.587)}{315}}$
$\mathrm{SE} \approx 0.028$

STEP 5

Determine the critical value $z$ for a $95\%$ confidence interval. Using a $z$ -distribution table, the critical value is:
$z \approx 1.96$

STEP 6

Calculate the margin of error $m$ :
$m = z \times \mathrm{SE} = 1.96 \times 0.028$
$m \approx 0.055$
The sample proportion is:
$\hat{p} = 0.587$
The standard error estimate is:
$\mathrm{SE} = 0.028$
The critical value is:
$z = 1.96$
The margin of error is:
$m = 0.055$

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Studdy Solution

STEP 1

1. The sample size n=315 n = 315 n=315.2. The number of successes x=185 x = 185 x=185.3. We are using a 95% 95\% 95% confidence interval.4. The population is large enough for the Central Limit Theorem to apply.

STEP 2

1. Calculate the sample proportion p^\hat{p}p^​.2. Calculate the standard error estimate SE\mathrm{SE}SE.3. Determine the critical value zzz for a 95%95\%95% confidence interval.4. Calculate the margin of error mmm.

STEP 3

Calculate the sample proportion p^\hat{p}p^​:p^=xn=185315\hat{p} = \frac{x}{n} = \frac{185}{315}p^​=nx​=315185​p^≈0.587\hat{p} \approx 0.587p^​≈0.587

STEP 4

Calculate the standard error estimate SE\mathrm{SE}SE:SE=p^(1−p^)n=0.587×(1−0.587)315\mathrm{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.587 \times (1 - 0.587)}{315}}SE=np^​(1−p^​)​​=3150.587×(1−0.587)​​SE≈0.028\mathrm{SE} \approx 0.028SE≈0.028

STEP 5

Determine the critical value zzz for a 95%95\%95% confidence interval. Using a zzz-distribution table, the critical value is:z≈1.96z \approx 1.96z≈1.96

STEP 6

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1. The sample size $n = 315$ .
2. The number of successes $x = 185$ .
3. We are using a $95\%$ confidence interval.
4. The population is large enough for the Central Limit Theorem to apply.

1. Calculate the sample proportion $\hat{p}$ .
2. Calculate the standard error estimate $\mathrm{SE}$ .
3. Determine the critical value $z$ for a $95\%$ confidence interval.
4. Calculate the margin of error $m$ .

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{185}{315}$
$\hat{p} \approx 0.587$

Calculate the standard error estimate $\mathrm{SE}$ :
$\mathrm{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.587 \times (1 - 0.587)}{315}}$
$\mathrm{SE} \approx 0.028$

Determine the critical value $z$ for a $95\%$ confidence interval. Using a $z$ -distribution table, the critical value is:
$z \approx 1.96$