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

Question

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:

$$\text{p-value}=P(X\ge 497\mid n=613,p=0.76)$$

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:

$$\text{p-value}=P(X\ge 497)=1-P(X<497)$$

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

$$P(X<497)=P(X\le 496)$$

Step 3: Calculate the exact p-value

We need to calculate $P(X\le 496)$. 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:

$$P(X\le 496)=P(X<497)$$

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

$$\text{p-value}=1-P(X\le 496)$$

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

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.