You wish to test the following claim () at a significance level of .

Question

     
     

You obtain a sample of size 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 is true) the probability of observing...
__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 = __________________

The p-value is...
__less than (or equal to)
__greater than

This test statistic leads to a decision to...
__reject the null
__accept the null
__fail to reject the null

As such, the final conclusion is that...
__There is sufficient evidence to warrant rejection of the claim that the population proportion is greater than 0.76.
__There is not sufficient evidence to warrant rejection of the claim that the population proportion is greater than 0.76.
__The sample data support the claim that the population proportion is greater than 0.76.
__There is not sufficient sample evidence to support the claim that the population proportion is greater than 0.76.

Answer 1

To conduct the hypothesis test and calculate the p-value, we need to define our null and alternative hypotheses based on the given claim.

Let's assume the null hypothesis \( H_0 \) is that the population proportion \( p \) is equal to 0.76. The alternative hypothesis \( H_a \) is that the population proportion \( p \) is greater than 0.76.

1. Define the null and alternative hypotheses:

   • \( H_0: p = 0.76 \)
   • \( H_a: p > 0.76 \)

2. Sample details:

   • Sample size \( n = N \) (please replace \( N \) with the actual size)
   • Number of successes \( x = 497 \)

3. Binomial Distribution: Since the binomial distribution is used, if \( H_0 \) is true, we will have a binomial distribution with parameters \( n \) and \( p = 0.76 \).

4. Calculate the p-value:

   • The p-value represents the probability of observing at least 497 successful observations when the population proportion is 0.76: \[ \text{p-value} = P(X \geq 497) = 1 - P(X < 497) \]
   • Here, \( P(X < 497) \) is the cumulative probability of getting fewer than 497 successes, which can be calculated using the binomial distribution.

You would use a binomial calculator or software to find this probability. Assuming you plug in the numbers correctly:

5. Interpret the p-value:

   • If the p-value is small (typically less than the significance level, say 0.05), that means we have evidence to reject \( H_0 \).

6. Decision:

   • If the p-value is less than or equal to the significance level, reject the null hypothesis.
   • If the p-value is greater than the significance level, fail to reject the null hypothesis.

7. Final conclusions: Based on the results:

   • If we reject \( H_0 \), it would imply that there is sufficient evidence to support the claim that the population proportion is greater than 0.76.
   • If we fail to reject \( H_0 \), it would imply that there is not sufficient evidence to support this claim.

Since I don't have the values for \( N \) (sample size), I cannot compute the exact p-value or make precise conclusions. Please provide the sample size \( n \) (the total number of observations), and I can help you further with the calculations.