Question

A biased coin was thrown 300 times and the tail turned up 120 times, then the standard error of the observed proportion of tails is

• A. \( 0.028 \)
• B. \( 0.28 \)
• C. \( 0.82 \)
• D. \( 0.082 \)

Hint:

The standard error of a proportion is a measure of the variability or uncertainty in the sample proportion as an estimate of the true population proportion. It is distinct from the standard deviation of the population. When dealing with sample proportions, use $\hat{p}$ and $\hat{q}$ in the formula $\sqrt{\frac{\hat{p}\hat{q}}{n}}$. If the problem mentioned a theoretical or population proportion $p$ (e.g., for an unbiased coin $p=0.5$), then you would use $p$ and $q$ (where $q=1-p$) in the formula $\sqrt{\frac{pq}{n}}$.

Answer 1

The Correct Option is A

Let $n$ be the number of trials (coin throws) and $x$ be the number of successes (tails turning up). Given: Total number of throws, $n = 300$. Number of tails, $x = 120$.
Step 1: Calculate the observed proportion of tails. The observed proportion of tails, denoted by $\hat{p}$, is given by: $$ \hat{p} = \frac{\text{Number of tails}}{\text{Total number of throws}} = \frac{x}{n} $$ $$ \hat{p} = \frac{120}{300} = \frac{12}{30} = \frac{2}{5} = 0.4 $$
Step 2: Calculate the proportion of heads (failure). The proportion of heads, denoted by $\hat{q}$, is given by: $$ \hat{q} = 1 - \hat{p} = 1 - 0.4 = 0.6 $$
Step 3: Calculate the standard error of the observed proportion. The standard error of the observed proportion ($\text{SE}_{\hat{p}}$) for a proportion is given by the formula: $$ \text{SE}_{\hat{p}} = \sqrt{\frac{\hat{p}\hat{q}}{n}} $$ Substitute the values of $\hat{p}$, $\hat{q}$, and $n$: $$ \text{SE}_{\hat{p}} = \sqrt{\frac{0.4 \times 0.6}{300}} $$ $$ \text{SE}_{\hat{p}} = \sqrt{\frac{0.24}{300}} $$ $$ \text{SE}_{\hat{p}} = \sqrt{0.0008} $$
Step 4: Calculate the square root and round to three decimal places. $$ \text{SE}_{\hat{p}} \approx 0.028284 $$ Rounding to three decimal places, we get $0.028$.