Question

NO LINKS!! Please help me with this probability 1c​

Answer 1

`\textsf{a)} \quad 78.67\% \pm 9.46\%`

or d) None of the answers are correct. (Please see notes below).

Step-by-step explanation:

P-hat is the probability that a given outcome will occur given a specified sample size.

`\boxed{\begin{minipage}{7.5 cm}\underline{P-hat formula}\\\\$\hat{p}=(X)/(n)$\\\\where:\\\phantom{ww}$\bullet$ $\hat{p}$ is the probability. \\ \phantom{ww}$\bullet$ $X$ is the number of occurrences of an event. \\ \phantom{ww}$\bullet$ $n$ is the sample size.\\\end{minipage}}`

Given:

• X = 59
• n = 75

Substitute the given values into the formula to find p-hat:

`\implies \hat{p}=(59)/(75) =0.78666666...`

The critical value for a 95% confidence level using normal distribution is:

`z=1.9600\;\;\sf (4\;d.p.)`

`\boxed{\begin{minipage}{5.2 cm}\underline{Margin of error}\\\\$ME=z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$\\\\where:\\\phantom{ww}$\bullet$ $z$ is the critical value. \\ \phantom{ww}$\bullet$ $\hat{p}$ is the sample proportion. \\ \phantom{ww}$\bullet$ $n$ is the sample size.\\\end{minipage}}`

Substitute the given values into the margin of error formula:

`\implies ME=1.9600 \cdot \sqrt{((59)/(75)\left(1-(59)/(75)\right))/(75)}`

`\implies ME=0.092715036...`

Therefore, an estimate of the proportion of young-adult novels that include a love triangle (including a margin of error) is:

`\implies \hat{p}\pm ME`

`\implies 0.7867\pm 0.0927`

`\implies 78.67\% \pm 9.27\%`

This result if not given in the list of answer options. However, the final result depends on the accuracy of the z-score and p-hat used in the calculations.

If we round the critical value to the nearest integer then:

`\implies z=2`

Substitute the rounded z-value into the margin of error formula:

`\implies ME=2\cdot \sqrt{((59)/(75)\left(1-(59)/(75)\right))/(75)}`

`\implies ME=0.094607180...`

Therefore, an estimate of the proportion of young-adult novels that include a love triangle (including a margin of error) using z = 2 is:

`\implies 78.67\% \pm 9.46\%`

Note: It is likely that this question required the critical value to be rounded to the nearest integer, however please note that this is not normal practice as it produces a different result, as evidenced above.

Answer 2

Answer: Choice A)
`\boldsymbol{78.67\% \ \pm \ 9.46\%}`

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Step-by-step explanation:

phat = 59/75 = 0.7867 = 78.67% approximately is the sample proportion

n = 75 is the sample size

Your teacher doesn't mention a confidence level, so I'll assume it's the default 95%.

Let's compute the margin of error at 95% confidence

E = z*sqrt(phat*(1-phat)/n)

E = 1.96*sqrt(0.7867*(1-0.7867)/75)

E = 0.0927 approximately

E = 9.27% approximately

Unfortunately the margin of error 9.27% isn't listed among the four answer choices, but let's change the z = 1.96 to z = 2 instead.

Recalculate the margin of error.

E = z*sqrt(phat*(1-phat)/n)

E = 2*sqrt(0.7867*(1-0.7867)/75)

E = 0.0946 approximately

E = 9.46% approximately

This margin of error is listed among the four answer choices.

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We found that

• phat = 78.67% approximately
• E = 9.46% approximately

The confidence interval in the format
`\text{phat} \ \pm \ \text{E}` is approximately
`78.67\% \ \pm \ 9.46\%` which points us to choice A as the answer.