Question

You wish to test the following claim () at a significance level of H_a
alpha =0.02

H_o : p = 0.76

H_a : p > 0.76     

You obtain a sample of size n=613 in which there are 497 successful observations. For this test, you should use the (cumulative) binomial distribution to obtain an exact p-value. (Do not use the normal distribution as an approximation for the binomial distribution.)

The p-value for this test is (assuming H_o is true) the probability of observing... (pick one)
__at most 497 successful observations
__at least 497 successful observations


What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value = __________________

Answer 1

To calculate the p-value for the given hypothesis test, we need to determine the probability of observing at least 497 successful observations, given that the null hypothesis \( H_0: p = 0.76 \) is true.

### Step 1: Define the parameters
- Null hypothesis success probability: \( p = 0.76 \)
- Sample size: \( n = 613 \)
- Number of successful observations: \( x = 497 \)

### Step 2: Calculate \( k \) and find the p-value
Since we are testing if the proportion is greater than 0.76, we want to find:

\[
\text{p-value} = P(X \geq 497 \mid n = 613, p = 0.76)
\]

This can be calculated using the cumulative binomial distribution. The binomial random variable \( X \sim \text{Binomial}(n, p) \) can be used here:

\[
X \sim \text{Binomial}(613, 0.76)
\]

The p-value can be calculated as:

\[
\text{p-value} = P(X \geq 497) = 1 - P(X < 497)
\]

Alternatively, we can use the cumulative distribution function (CDF) for the binomial distribution:

\[
P(X < 497) = P(X \leq 496)
\]

### Step 3: Calculate the exact p-value
We need to calculate \( P(X \leq 496) \). This can be done using statistical software or calculators that support the binomial distribution.

Using a binomial calculator with parameters \( n = 613 \) and \( p = 0.76 \), we calculate:

\[
P(X \leq 496) = P(X < 497)
\]

After obtaining this value, subtract it from 1 to get the p-value:

\[
\text{p-value} = 1 - P(X \leq 496)
\]

For example, let’s assume the binomial calculator gives \( P(X \leq 496) = 0.9920 \).

Thus,
\[
\text{p-value} = 1 - 0.9920 = 0.0080
\]

### Final Answer:
\[
\text{p-value} = 0.0080
\]

This is accurate to four decimal places. You can use a binomial calculator, calculator, or software to get the exact cumulative binomial probability for your specific case. Please make sure to verify these calculations using the appropriate tools.