Math / Data & Statistics

QuestionLeft-Handed Lawyers Approximately $10 \%$ of Americans are left-handed (we will treat this as a known population parameter). A study on the relationship between handedness and profession found that in a random sample of 105 lawyers, 16 of them were lefthanded. ${ }^{1}$ Test the hypothesis that the proportion of left-handed lawyers differs from the proportion of left-handed Americans. ${ }^{1}$ Schachter, S. and Ransil, B., "Handedness Distributions in Nine Professional Groups," Perceptual and Motor Skills, 1996; 82: 51-63.
Part 1
Clearly state the null and alternative hypotheses. Let $p$ be the proportion of left-handed lawyers. eTextbook and Media Save for Later Attempts: 0 of 5 used Submit Answer
Part 2
Calculate the test statistic and $p$ -value. Round your answer for the test statistic to two decimal places, and your answer for the $p$ -value to three decimal places. test statistic $=$ $\square$ i $\square$ $p$ -value $=$ eTextbook and Media

Studdy Solution

STEP 1

1. The population proportion of left-handed Americans is $p_0 = 0.10$ .
2. The sample size of lawyers is $n = 105$ .
3. The number of left-handed lawyers in the sample is $x = 16$ .
4. We will use a significance level $\alpha$ to determine the $p$ -value.

STEP 2

1. State the null and alternative hypotheses.
2. Calculate the sample proportion.
3. Calculate the test statistic.
4. Determine the $p$ -value.

STEP 3

State the null and alternative hypotheses.
- Null Hypothesis ( $H_0$ ): The proportion of left-handed lawyers is equal to the proportion of left-handed Americans. $ H_0: p = 0.10 \]
- Alternative Hypothesis ( $H_a$ ): The proportion of left-handed lawyers differs from the proportion of left-handed Americans. $ H_a: p \neq 0.10 \]

STEP 4

Calculate the sample proportion ( $\hat{p}$ ).
$\hat{p} = \frac{x}{n} = \frac{16}{105} \approx 0.1524$

STEP 5

Calculate the test statistic using the formula for a proportion:
$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
Substitute the known values:
$z = \frac{0.1524 - 0.10}{\sqrt{\frac{0.10 \times 0.90}{105}}}$
Calculate the denominator:
$\sqrt{\frac{0.10 \times 0.90}{105}} \approx 0.0293$
Calculate the test statistic:
$z \approx \frac{0.1524 - 0.10}{0.0293} \approx 1.79$

STEP 6

Determine the $p$ -value for the two-tailed test using the standard normal distribution.
The $p$ -value is calculated as:
$p\text{-value} = 2 \times P(Z > |z|)$
Using a standard normal distribution table or calculator:
$P(Z > 1.79) \approx 0.0367$
Thus, the $p$ -value is:
$p\text{-value} = 2 \times 0.0367 \approx 0.0734$
The test statistic is approximately $1.79$ and the $p$ -value is approximately $0.073$ .

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