1. Bing is a student researcher trying to disprove
Bong’s claim that there are, on the average, 500
nerds in a box. A random sample of 30 boxes of
nerds are examined and the following counts were
observed. Test the claim of Bong that there are, on
the average, 500 nerds in a box at α = 0.01.

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.

Bing is a student researcher trying to disprove Bong’s claim that there are, on the average, 500 nerds in a box. A random sample of 30 boxes of nerds are examined and the following counts were observed. Test the claim of Bong that there are, on the average, 500 nerds in a box at α = 0.01.

Perform a hypothesis test on Bong's claim that the average number of nerds in a box is 500. This problem uses a two-tailed t-test with a significance level of α = 0.01, and covers statistical concepts like hypothesis testing, critical values, and the Central Limit Theorem.

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