A coin was flipped 60 times and came up heads 38 times. (a)at the .10 level of significance, is the coin biased towards heads? Show your decision rule and calculations. (b) calculate a p-value and interpret it.

Here might be one way to do this problem:

Null hypothesis is that the coin is fair. Ho: p = .5
Alternate hypothesis is that the coin is unfair. Ha: p not equal to .5

Using the binomial formula: P(x) = (nCx)(p^x)[q^(n - x)]
...where n = number of coin tosses, x = number of times came up heads, p = probability given in the null hypothesis, q = 1 - p.

Using your data:
P(60) = (60C38)(.5^38)(.5^22)
= .0123 (rounded to four decimal places)

If the alternate hypothesis uses "not equal to" then we multiply the results by 2. Therefore, 2 * .0123 = .0246
If the alternate hypothesis would have shown a specific direction, then we could have used .0123 as is.

Reject the null hypothesis if the test statistic above is less than .10 level of significance; otherwise, do not reject null.
The p-value is .0246 (the p-value is the actual level of the test statistic). You should be able to take it from here.

I hope this will help.

Correction (this does not change the outcome):

Using the binomial formula: P(x) = (nCx)(p^x)[q^(n - x)]
...where n = number of coin tosses, x = number of times came up heads, p = probability given in the null hypothesis, q = 1 - p.

Using your data:
P(38) = (60C38)(.5^38)(.5^22)
= .0123 (rounded to four decimal places)

*** Changed P(60) to P(38). Sorry for any confusion!

Thank you for your help.

To determine if the coin is biased towards heads, we need to perform a hypothesis test.

(a) Hypothesis Testing:
Null Hypothesis (H0): The coin is unbiased (probability of getting heads is 0.5)
Alternative Hypothesis (Ha): The coin is biased towards heads (probability of getting heads is greater than 0.5)

Using the binomial distribution, we can calculate the probability of getting heads 38 times out of 60 flips assuming the coin is unbiased. If this probability is very low, we can reject the null hypothesis and conclude that the coin is biased towards heads.

Now, let's calculate the critical value and the test statistic to make a decision:

1. Determine the critical value:
At the significance level of 0.10 (10%), that corresponds to a one-tailed test, as we are interested in whether the probability of getting heads is greater than 0.5. Looking up critical values in the z-table, for a significance level of 0.10, the critical value is approximately 1.28.

2. Calculate the test statistic:
To calculate the test statistic (z-score), we use the formula:
z = (X - μ) / √(p * (1 - p) / n),
where X is the number of heads, μ is the expected mean (np), p is the null probability (0.5), and n is the number of flips.

In this case, X = 38, μ = 60 * 0.5 = 30, p = 0.5, and n = 60.
Plugging the values into the formula:
z = (38 - 30) / √(0.5 * (1 - 0.5) / 60) ≈ 2.27

3. Make a decision:
Since the calculated test statistic (z) value of 2.27 is greater than the critical value of 1.28, we can reject the null hypothesis.

Therefore, at the 0.10 level of significance, we conclude that the coin is biased towards heads.

(b) Calculate the p-value:
The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. Since we are conducting a one-tailed test to see if the probability of getting heads is greater than 0.5, we need to find the area under the standard normal curve to the right of the calculated test statistic value (z = 2.27).

Looking up the p-value in the z-table, we find that the area to the right of z = 2.27 is approximately 0.011.

Interpretation of the p-value:
The p-value of 0.011 means that there is an approximately 1.1% chance of observing a result as extreme as, or more extreme than, getting 38 heads in 60 flips, assuming the coin is unbiased.

In conclusion, the p-value of 0.011 is less than the significance level of 0.10, indicating strong evidence to reject the null hypothesis. We can interpret this as evidence that the coin is biased towards heads.