Question

A coin was flipped 60 times and came up heads 38 times. At the .10 level of significance, is the coin biased toward heads?

(a-1) H₀: ππ ≤ .50 versus H₁: ππ > .50. Choose the appropriate decision rule at the .10 level of significance.

1. Reject H₀ if z > 1.282
2. Reject H₀ if z < 1.282

(a-2) Calculate the test statistic.

Answer 1

Let  X = Event that head comes up on tossing a coin; Given that sample size = n = 60 and x = 38.

Let denote the probability of success, We have to test:

H0: π ≤ .50 versus H1: π > .50

Given,

Let denote the sample proportion, obtained using the formula:

The appropriate statistical test to test the above hypothesis would be a One sample test for Proportion.But before running the test we must first ensure whether all the assumptions of the test are satisfied. The assumptions of One sample test for Proportion are as follows:

· The data is randomly collected from the population

· The underlying distribution of the population is binomial distribution.

· There are only two possible outcomes in each trial: Heads or Tails

· When np ≥ 5 and np(1 – p) ≥ 5, the binomial distribution can be approximated by the normal distribution.

Assumpting that all the assumptions of the test are satisfied, we may go for computing the test statistic, given by the formula:

with critical / rejection region given by since, this is a right tailed test i.e. extreme values to the right of the curve would provide strong evidence to reject the null ( H1: π > .50 )

For , looking for area 0.90 in standard normal table:

To obtain the exact critical value, we may use the excel function:

We get the critical value Z = 1.282.

Hence, from the critical region defined above,

We may reject H0 if Z > 1.282

Substituting the given values in the test statistic,

= 2.066

The test statistic is obtained as Z = 2.066