To test Bong’s claim that there are, on average, 500 nerds in a box using Bing’s sample data, we will perform a hypothesis test. Since we are testing the average count, this will be a one-sample t-test (if the population standard deviation is unknown) or a z-test (if the population standard deviation is known).

Here are the steps to perform the hypothesis test:

Step 1: Set up the hypotheses

• Null Hypothesis (H₀): The average number of nerds in a box is 500. $H₀: \mu = 500$
• Alternative Hypothesis (H₁): The average number of nerds in a box is not 500. $H₁: \mu \neq 500$

This will be a two-tailed test since we are checking if the average is not equal to 500.

Step 2: Identify the significance level (α)

The significance level is given as α = 0.01.

Step 3: Gather the sample data

You have mentioned a random sample of 30 boxes, but the actual counts of nerds in each of the 30 boxes are not provided. To proceed with the test, I need the observed counts from the sample to calculate the sample mean ($\bar{x}$) and the sample standard deviation ($s$).

Step 4: Calculate the test statistic

Once the data is available, we will calculate the test statistic. If the population standard deviation is unknown, the formula for the t-statistic is:

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ Where:

• $\bar{x}$ = sample mean
• $s$ = sample standard deviation
• $n$ = sample size (30 in this case)
• $\mu$ = population mean (500)

Step 5: Determine the critical value and make a decision

For a two-tailed t-test with a significance level of 0.01 and $n - 1 = 29$ degrees of freedom, we will use the t-distribution to find the critical value $t_{\alpha/2}$. Compare the calculated t-statistic with the critical value:

• If $|t| > t_{\alpha/2}$, reject the null hypothesis $H₀$.
• Otherwise, fail to reject $H₀$.

If you provide the observed counts of nerds in the 30 boxes, I can help you calculate the necessary values and complete the test. Do you have the sample data?

Follow-up Questions:

1. What are the actual counts of nerds in the 30 boxes?
2. Is the population standard deviation known, or should we proceed with the t-test?
3. Are there any other assumptions to consider, like whether the data follows a normal distribution?
4. Would you like to see the critical value calculations based on your degrees of freedom?
5. Should we calculate a confidence interval for the mean number of nerds in a box?

Tip: When conducting a hypothesis test, always visualize the problem with a distribution curve to better understand the critical regions.