Complete Question
Your PI claims that the proportion of Morpho butterflies in a population that are blue is 0.3 .A sample is independently obtained from this population. Of 200 sampled Morphos, 50 turn out to be blue.Given only this information, carry out a hypothesis test to evaluate the claim. What is closest to the p-value that you obtain?
A 0.019
B 0.038
C 0.070
D 0.139
Answer:
The correct answer is D
Step-by-step explanation:
From the question we are told that
The population proportion of blue butterflies is [tex]p = 0.3[/tex]
The sample size is [tex]n = 200[/tex]
The sample mean is [tex]\= x = 50[/tex]
The Null Hypothesis is mathematically represented as
[tex]H_o : p = 0.3[/tex]
The Alternative Hypothesis is mathematically represented as
[tex]H_a : p \ne 0.3[/tex]
Now the sample proportion is mathematically represented as
[tex]\r p = \frac{\= x}{n}[/tex]
substituting values
[tex]\r p = \frac{50 }{200 }[/tex]
[tex]\r p = 0.25[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{\r p - p }{\sqrt{\frac{p(1-p)}{n } } }[/tex]
substituting values
[tex]z = \frac{ 0.25 - 0.3 }{\sqrt{\frac{0.3(1-0.3)}{200 } } }[/tex]
[tex]z = -1.54[/tex]
The p-value for a two-tailed test is mathematically represented as
for lower -tail test
[tex]p-value = P(Z \le z | H0\ is \ true) = cdf(z )[/tex]
for higher-tail test
[tex]p-value = P(Z \ge z | H0\ is \ true) = 1- cdf(z )[/tex]
for this test i assumed a 0.05 level of significance
Now
[tex]cdf(z)[/tex] is the cumulative distribution function for test statistics under the null hypothesis
Which can be calculated using MInitab (A statistics calculator )
for lower-tail test
The p-value is not significant
for higher-tail test
p-value is
[tex]1- cdf(-1.54) = 0.125[/tex]
