When conducting a hypothesis test with a binomial distribution (sometimes called a Binomial Test), there are three ways to calculate the P-value (with additional variations possible). The only exact calculation is to use the binomial probability distribution. The other methods are approximations using the standardized normal distribution (when certain criteria have been achieved). Of these two methods, one can use the sample counts or one can use the sample proportions. Furthermore, it is possible in both of these approximating cases to apply a continuity correction to account for the use of a continuous distribution to approximate a discrete distribution.

Question

This problem introduces the method to obtain an exact P-value using the binomial distribution.  Also, this method uses Excel to obtain the answers.

For this demonstration problem, we will test a hypothesis that a population proportion has increased since the last time it was measured.  Previously, the population proportion was measured at 18%.  For the current analysis, a sample of n=100 randomly chosen subjects was obtained, and 23 of those demonstrated the observation of interest (i.e., a success).

To start, we clearly construct the hypotheses for this problem.  Because the researcher suggested the proportion has increased, this would suggest a one-tailed test (as can be seen in the choice of H_a`):

H_o : p = 0.18
H_a : p > 0.18

Using the binomial distribution, the test statistic from the sample would simply be the sample count (the number of successful observations): k = 23

The P-value for this scenario would be observing this count or one more extreme.  With the alternative hypothesis suggesting that values at or below 18% would be unsurprising, this would suggest that the observed count or larger would constitute the potentially extreme responses.  Thus, the P-value would be: P(X >= 23 | p=0.18, n=100)

Using Excel, this value can be calculated exactly:
=1-BINOMDIST(22,100,18%,TRUE)

which should return a P-value of 0.1223.  Thus, with a traditional significance level of either or , this P-value would result in failing to reject the null hypothesis.  Thus, there is not enough sample evidence to support the claim that the population proportion has increased.

Note:  For a two-tailed hypothesis test, the calculation can become a bit tricky.  In this case, it is necessary to take the counts at and above the sample count, but it is also necessary to determine comparable counts below the hypothesized population count.  This would be obtained using the formula

n*p - (k-n*p)

  
This value is the count below the hypothesized mean count by the same distance as the sample count was above the mean count.  For this demonstration example, this value would be 13.  To obtain the P-value from Excel, you would use the following formula:
=BINOMDIST(13,100,18%,TRUE)+1-BINOMDIST(22,100,18%,TRUE)
Though, it is recommended that the two-tailed test only be used with one of the approximation methods as the calculations are less cumbersome.

Answer 1

To summarize the hypothesis testing process you described using the binomial distribution, let's break it down step-by-step:

Step 1: State the Hypotheses

For this example, we are testing whether the population proportion has increased from a previous measurement of 18%. Thus, the hypotheses can be stated as follows:

• Null Hypothesis (H₀): \( p = 0.18 \) (the population proportion has not increased)
• Alternative Hypothesis (Hₐ): \( p > 0.18 \) (the population proportion has increased)

Step 2: Collect the Sample Data

• Sample size \( n = 100 \)
• Number of successes \( k = 23 \)

Step 3: Define the Test Statistic

The test statistic in a binomial test is the observed count of successes \( k \):

• \( k = 23 \)

Step 4: Calculate the P-Value

Since the alternative hypothesis is one-tailed (we are testing if the proportion has increased), the P-value will be calculated as follows:

• We need to find \( P(X \geq 23 | p = 0.18, n = 100) \).

In Excel, the exact calculation for the P-value can be performed using:

=1 - BINOMDIST(22, 100, 0.18, TRUE)

This calculates the cumulative probability of getting up to \( 22 \) successes and subtracts it from \( 1 \) to find the probability of getting \( 23 \) or more successes.

• Result: The P-value returned from this calculation is approximately \( 0.1223 \).

Step 5: Decision Rule

With a conventional significance level (α) of \( 0.05 \) or \( 0.01 \):

• If P-value ≤ α, reject the null hypothesis.
• If P-value > α, fail to reject the null hypothesis.

In this case:

• Since \( 0.1223 \) is greater than both \( 0.05 \) and \( 0.01 \), we fail to reject the null hypothesis.

Step 6: Conclusion

There is insufficient evidence to support the claim that the population proportion has increased from 18%.

Step 7: Conducting a Two-Tailed Test (optional)

While the one-tailed test is adequate for this scenario, if we were to conduct a two-tailed test, we would:

1. Calculate the lower bound for the count based on the hypothesized proportion. If the observed count is 23, the equivalent lower bound count should be: \[ n \cdot p - (k - n \cdot p) = 100 \cdot 0.18 - (23 - 18) = 18 - 5 = 13 \] So the lower bound is \( 13 \).

2. Use the following formula in Excel to compute the two-tailed P-value:

=BINOMDIST(13, 100, 0.18, TRUE) + (1 - BINOMDIST(22, 100, 0.18, TRUE))

This would give you the total P-value for the two-tailed test, taking into account both the extreme counts above \( 22 \) and below \( 13 \).

Final Note

It's more common to use approximation methods for two-tailed tests due to ease of computation, but for exact calculations, the binomial probabilities are precisely implemented in the manner described above. For one-tailed tests, direct calculations using the binomial distribution provide a straightforward method to determine the P-value.