Question

A person claims to be able to make make a fair coin land heads more often than expected, using the power of telekenesis. A skeptical scientist investigates this claim and tosses a fair coin 75 times. It lands heads 41 times. Is there evidence to support the person’s claims?

(a) State a sensible null hypothesis

(b) State the precise definition of p-value and explain what “more extreme” means in this context

(c) Is a one-sided or two-sided test needed? justify

(d) Using dbinom(), or otherwise, calculate a p-value and interpret.

(e) (harder) Calculate a p-value using the normal approximation and compare your answer with the exact binomial p-value above.

Notes:
• Show detailed working, including appropriate mathematical notation for each question. For most questions this will involve showing your working from R or R Studio, (e.g. cut-and-paste commands and output from an R session).

• Any question involving regression will score 0 marks unless a scattergraph is produced.

• No Additional Info provided

Answer 1

(a) The hypothesis being tested is:

H₀: p = 0.5

H_a: p > 0.5

(b) In statistical hypothesis testing, the p-value or probability value is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct.

(c) This is a one-sided test because we have a direction.

(d) 0.09222417

(e) p-value = 0.2094617

Since the p-value (0.2094617) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that p > 0.5.

The R code is:

dbinom(41,75,41/75)
x = 41
n = 75
mu = 75*0.5
sd = sqrt(75*0.5*0.5)
z = (x - mu)/sd
pnorm(z, lower.tail = FALSE)