Question

A student wants to determine if pennies are really fair, meaning equally likely to land heads up or tails up. He flips a random sample of 50 pennies and finds that 28 of them land heads up. Calculate the p-value and state the conclusion. Use α = .05.

Answer 1

To determine if pennies are fair, you can use a hypothesis test to compare the observed results to what would be expected if the pennies were fair. Calculate the p-value by finding the probability of getting the observed result or a more extreme result, assuming the null hypothesis is true. Compare the p-value to the significance level (α) to make a conclusion.

Step-by-step explanation:

We can use a binomial test to calculate the p-value, which is the probability of observing the given result if the null hypothesis is true. Using a two-tailed test, we calculate the probability of getting 28 or more heads out of 50 flips assuming the null hypothesis is true. This probability is the p-value.

To calculate the p-value, we first find the probability of getting exactly 28, 29, ..., 50 heads out of 50 flips assuming the null hypothesis is true. We then sum up these individual probabilities to get the final p-value.

1. Calculate the individual probabilities for getting 28, 29, ..., 50 heads out of 50 flips using the binomial probability formula.
2. Sum up these individual probabilities to get the final p-value.

After calculating the p-value, we compare it to the significance level (α) of 0.05. If the p-value is less than α (0.05), we reject the null hypothesis and conclude that the pennies are not fair. If the p-value is greater than or equal to α (0.05), we fail to reject the null hypothesis and do not have enough evidence to conclude that the pennies are not fair.

Answer 2

Pennies are not really fair.

Step-by-step explanation:

Given that a student wants to determine if pennies are really fair, meaning equally likely to land heads up or tails up.

Let us do hypothesis test claiming p = 0.50

`H_0: p =0.50\\H_a: p \\eq 0.50`

(two tailed test)

p = proportion of heads to total

Sample proportion P =
`(28)/(50) \\=0.56`

Std error =
`\sqrt{(pq)/(n) } =\sqrt{(0.25)/(50) } \\=0.0707`

Test statistic = p diff/std error

=
`(0.5)/(0.0707) \\=7.07`

p value = 0

Since p is less than 0.05

reject H0.

It is statistically evident that the events are not equally likely for head or tail.