The accompanying table contains data on the​ weight, in​ grams, of a sample of 50 tea bags produced during an​ eight-hour shift.   5.63 5.46 5.41 5.39 5.54 5.36 5.56 5.45 5.52 5.41 5.57 5.41 5.54 5.56 5.55 5.64 5.54 5.44 5.46 5.52 5.47 5.39 5.48 5.62 5.52 5.31 5.68 5.28 5.49 5.53 5.75 5.58 5.41 5.56 5.59 5.49 5.31 5.49 5.52 5.58 5.62 5.46 5.42 5.26 5.57 5.61 5.52 5.58 5.69 5.35.       State the null and alternative hypothesis, p-value and Construct a 95​% confidence interval estimate of the population mean amount of tea per bag.

Let's break this down step by step. We are asked to:

1. **State the null and alternative hypothesis**
2. **Find the p-value**
3. **Construct a 95% confidence interval estimate of the population mean weight of tea per bag**

### 1. **State the Null and Alternative Hypothesis**

- **Null Hypothesis (H₀):** The population mean weight of the tea bags is equal to a specified value (e.g., 5.50 grams).
  
  \( H_0: \mu = 5.50 \)

- **Alternative Hypothesis (H₁):** The population mean weight of the tea bags is different from 5.50 grams.
  
  \( H_1: \mu \neq 5.50 \)

This is a two-tailed test since we are checking if the mean is either higher or lower than 5.50 grams.

### 2. **Find the p-value**

We will use a **t-test** to compute the p-value, as we are dealing with a sample (n = 50) rather than the whole population, and the population standard deviation is unknown. The steps are:

- First, we need to calculate the sample mean (\(\bar{x}\)) and sample standard deviation (s).
- Then, using a t-distribution, we compute the t-statistic based on the null hypothesis.
- Finally, we compute the p-value.

I will perform these calculations now.### 2. **p-value:**
- The sample mean (\(\bar{x}\)) is **5.5018 grams**.
- The sample standard deviation (s) is **0.1071 grams**.
- The calculated t-statistic is **0.119**.
- The corresponding **p-value** is **0.906**.

Since the p-value (0.906) is much larger than the typical significance level (\(\alpha = 0.05\)), we **fail to reject the null hypothesis**. This means that there is insufficient evidence to conclude that the population mean weight of tea bags is different from 5.50 grams.

### 3. **95% Confidence Interval:**
The 95% confidence interval for the population mean weight of the tea bags is:
\[
(5.4714, 5.5322) \, \text{grams}
\]

This means we are 95% confident that the true population mean lies between **5.4714 grams** and **5.5322 grams**.

### Summary:
- **Null Hypothesis (H₀):** \(\mu = 5.50\)
- **Alternative Hypothesis (H₁):** \(\mu \neq 5.50\)
- **p-value:** 0.906 (Fail to reject the null hypothesis)
- **95% Confidence Interval:** (5.4714, 5.5322) grams

Would you like further details on any step? Here are some related questions you might consider:

1. What is the importance of the sample size in hypothesis testing?
2. How does a two-tailed test differ from a one-tailed test?
3. What assumptions must be met to perform a t-test?
4. How would the confidence interval change if we used a different confidence level?
5. Can you explain the meaning of the p-value in this context?

**Tip:** The confidence interval provides a range where the true population mean is likely to lie, while the p-value tells us whether the sample data provide strong evidence against the null hypothesis.

The accompanying table contains data on the​ weight, in​ grams, of a sample of 50 tea bags produced during an​ eight-hour shift. 5.63 5.46 5.41 5.39 5.54 5.36 5.56 5.45 5.52 5.41 5.57 5.41 5.54 5.56 5.55 5.64 5.54 5.44 5.46 5.52 5.47 5.39 5.48 5.62 5.52 5.31 5.68 5.28 5.49 5.53 5.75 5.58 5.41 5.56 5.59 5.49 5.31 5.49 5.52 5.58 5.62 5.46 5.42 5.26 5.57 5.61 5.52 5.58 5.69 5.35. State the null and alternative hypothesis, p-value and Construct a 95​% confidence interval estimate of the population mean amount of tea per bag.

Learn how to perform hypothesis testing for the mean weight of tea bags produced during an eight-hour shift. This problem involves calculating the p-value and constructing a 95% confidence interval estimate based on sample data. Perfect for Statistics 101 students!

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