Question

(1 point) A noted psychic was tested for ESP. The psychic was presented with 180 cards face down and was asked to determine if the card was one of 5 symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 43 cases. Let pp represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 180 trials can be treated as an SRS from the population of all guesses. To see if there is evidence that the psychic is doing better than just guessing, we test H0:p=.2H0:p=.2 Ha:p>.2Ha:p>.2 (a) What is the zz-statistic for this test? (b) What is the P-value of the test?

Answer 1

Answer:

a) $z=\frac{0.239 -0.2}{\sqrt{\frac{0.2(1-0.2)}{180}}}=1.308$  

b) $p_v =P(z>1.308)=0.0954$  

Step-by-step explanation:

Data given and notation

n=180 represent the random sample taken

X=43 represent the correct guesses

$\hat p=\frac{43}{180}=0.239$ estimated proportion correct guesses

$p_o=0.2$ is the value that we want to test

$\alpha$ represent the significance level

z would represent the statistic (variable of interest)

$p_v$ represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the ture proportion is higher than 0.2 or no.:  

Null hypothesis:$p=0.2$  

Alternative hypothesis:$p > 0.2$  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

$z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}$ (1)  

The One-Sample Proportion Test is used to assess whether a population proportion $\hat p$ is significantly different from a hypothesized value $p_o$.

Part a: Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

$z=\frac{0.239 -0.2}{\sqrt{\frac{0.2(1-0.2)}{180}}}=1.308$  

Part b: Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

$p_v =P(z>1.308)=0.0954$