Flower Harmony, a renowned flower shop, takes pride in its exquisite floral arrangements. Among their best-selling products are the Eternal Roses—preserved roses that defy time, lasting for months without wilting. Recently, Linda Lee, the shop owner, noticed fluctuations in the freshness of the Eternal Roses delivered by a key supplier. Suspecting that the average humidity level during transportation might be affecting the quality of the roses, the owner decides to investigate it further.

Drawing from historical data during periods when the roses’ freshness was consistently considered good, the owner found that the average humidity level during transportation hovered around 65.1%. With this information, she aims to investigate whether the current average humidity level deviates significantly from this benchmark.

Linda collects data on the humidity levels during the last fifteen shipments of Eternal Roses from that supplier. Each shipment consists of multiple boxes of roses, and she records the average humidity level for each shipment:

Table 1

Shipment #	Average Humidity Level (%)
1	65.1
2	69.1
3	59.0
4	68.3
5	66.3
6	69.3
7	71.7
8	68.3
9	72.2
10	67.3
11	70.9
12	66.3
13	69.2
14	73.8
15	68.2

What is the p-value of the hypothesis ?

To determine the p-value of the hypothesis, we'll perform a one-sample t-test. This test will help us determine whether the mean humidity level during the last fifteen shipments is significantly different from the historical benchmark of 65.1%.

### Steps to perform the t-test:

1. **State the hypotheses:**
   - Null Hypothesis (\(H_0\)): The average humidity level during transportation is equal to the benchmark, i.e., \(\mu = 65.1\%\).
   - Alternative Hypothesis (\(H_1\)): The average humidity level during transportation is different from the benchmark, i.e., \(\mu \neq 65.1\%\).

2. **Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (s):**
   - The sample consists of 15 humidity measurements.
   - We need to compute the sample mean and sample standard deviation.

3. **Calculate the t-statistic:**
   \[
   t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
   \]
   Where:
   - \(\bar{x}\) = sample mean
   - \(\mu_0\) = population mean (65.1%)
   - \(s\) = sample standard deviation
   - \(n\) = sample size (15)

4. **Determine the degrees of freedom (df):**
   - \(df = n - 1\)

5. **Calculate the p-value:**
   - Using the t-distribution with \(df = n-1\), find the p-value associated with the calculated t-statistic.

Let's proceed with the calculations. I'll start by computing the sample mean and standard deviation.The calculations yield the following results:

- **Sample Mean (\(\bar{x}\))**: 68.33%
- **Sample Standard Deviation (s)**: 3.50%
- **t-Statistic (t)**: 3.57
- **p-Value**: 0.0031

### Interpretation:
The p-value is 0.0031, which is very small and less than the common significance level of 0.05. This suggests that the average humidity level during the last fifteen shipments is significantly different from the historical benchmark of 65.1%. Therefore, Linda has strong evidence to reject the null hypothesis.

Would you like more details or have any questions?

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**Relative Questions:**
1. How do you interpret the p-value in the context of this test?
2. What would it mean if the p-value were much higher, say 0.2?
3. Can the sample size affect the result of the t-test?
4. How does the t-statistic relate to the p-value?
5. What assumptions must be met to perform a valid t-test?
6. What other tests could be used to validate these findings?
7. What are the implications for Linda's business based on this result?
8. How would you explain this finding to someone without a statistical background?

**Tip:** When conducting a hypothesis test, always ensure your sample data meets the assumptions of the test to avoid misleading results.

Learn how to calculate the p-value for hypothesis testing using t-tests with the example of Eternal Roses' humidity levels. Understand statistical concepts such as t-tests, p-values, and hypothesis testing. Detailed solution and interpretation provided.

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