Question

More lefties?In the population of people in the United States, about 10% are left-handed. After bumping elbows at lunch with several left-handed students, Simon wondered if more than $10\%$of students at his school are left-handed. To investigate, he selected an SRS of $50$students and found $8$lefties $(\stackrel{⏜}{p}=8/50=0.16)$.

To determine if these data provide convincing evidence that more than $10\%$of the students at Simon’s school are left-handed, $200$trials of a simulation were conducted. Each dot in the graph shows the proportion of students that are left-handed in a random sample of $50$students, assuming that each student has a $10\%$chance of being left handed.

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

b. Use the simulation results to estimate the P-value of the test in part (a). Interpret the $P$-value.

c. What conclusion would you make?

Short answer

a. $H_0:p=10\%=0.10$and $H_1:p>0.01$

b. There is $12\%$chances of getting at least $8$left handed students in sample of $50$when proportion for the same is $0.10$.

c. No convincing evidence that more than $10\%$of students at school are left handed.

Step-by-step solution

Given Information

It is given that Claim is more than $10\%$proportion.

$\hat{p}=8/50=0.16$

Appropriate hypothesis

Population value is equal to value given in claim is null hypothesis.

$H_0:p=10\%=0.10$

Claim can be null or alternate hypothesis.

Null: Population proportion is equal to value in claim.

In it is claim, then alternate hypothesis is opposite of null hypothesis.

$H_1:p>0.01$

$p$ is population proportion of all left handed students.

Step 3: P Value

From part (a) $H_0:p=10\%=0.10$

$H_1:p>0.10$

From diagram, it is observed that $\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$200$}\right.$dots are $0.16$or right of it.

Hence,$\text{P-value}=\frac{24}{200}=\frac{3}{25}=0.12=12\%$

Conclusion

If $P\mathrm{value}<\mathrm{\alpha }$, null hypothesis is rejected.

Commonly $\alpha =0.01,0.05,0.10$, hence $P$value is greater than all three. Hence it does not reject null hypothesis.

There is no enough evidence that more than $10\%$students are left handed.