It is commonly said that 10 of people are left-handed, but Lilianna suspected that a higher proportion of art students at her university are left-handed. To test this theory, she took a sample of 150 art students and found that p^=14% of the sample was left-handed.
To see how likely a sample like this was to happen by random chance alone, Lilianna performed a simulation. She took a sample of n=150 students from a population where 10% of the students were left-handed, and she recorded what proportion of the sample was left-handed. She repeated this process for a total of 50 samples. Here are the sample proportions from her 50 samples:

She wants to test H0:p=10% vs. Ha:p>10% where p is the proportion of art students at her university who are left-handed.

Based on these simulated results, what is the approximate p-value of the test?
Note: The sample result was p^=14%.

Answer: p-value ≈ 0.08

Step-by-step explanation:

The approximate p-value of the test is 0.0136 as per the data given in question.

The p value is a number calculated from a statistical test that describes the likelihood of finding a specific set of observations if the null hypothesis is true.

To find the approximate p-value of the test, we need to compare the sample proportion p^ to the hypothesized proportion p0 = 10% under the null hypothesis H0:p=10% vs. Ha:p>10%.

Since the sample size n is large (n = 150) and the null hypothesis assumes a normal distribution.

We can use the normal approximation to the binomial distribution. Under the null hypothesis, the test statistic z is given by:

z = (p^ - p0) / sqrt(p0 * (1 - p0) / n)

Plugging in the values, we get:

z = (0.14 - 0.1) / sqrt(0.1 * 0.9 / 150)

z = 2.22

The approximate p-value for this one-tailed test can be found by looking up the area to the right of z = 2.22 under the standard normal distribution.

Using a standard normal distribution table or calculator, we find that the area to the right of z = 2.22 is approximately 0.0136.

Therefore, the approximate p-value of the test is 0.0136. This means that if the true proportion of left-handed art students at the university is 10%, there is approximately a 1.36% chance of getting a sample proportion of 14% or higher by random chance alone.

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