A few days ago, Melissa adjusted the price of Eternal Roses. Now, she wants to assess whether this price change has affected the daily sales. Table 2 shows Eternal Roses sales data for the first 10 days after the price change.

Table 2

Day	Eternal Roses Sales (in units)
1	93
2	117
3	78
4	130
5	122
6	117
7	98
8	102
9	91
10	101

Calculate the 90% confidence interval for the true mean of daily Eternal Roses sales using the data sample collected.

What is the lower limit of this 90% confidence interval?
Enter your answer with two decimal places

To calculate the 90% confidence interval for the true mean of daily Eternal Roses sales, we'll follow these steps:

1. **Calculate the sample mean (\(\bar{x}\)):**

\[
\bar{x} = \frac{\text{Sum of all sales}}{\text{Number of days}}
\]

2. **Calculate the sample standard deviation (s):**

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]

3. **Determine the t-score for a 90% confidence interval with \(n - 1\) degrees of freedom.**

4. **Calculate the margin of error (ME):**

\[
\text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

5. **Determine the confidence interval:**

\[
\text{CI} = \left(\bar{x} - \text{ME}, \bar{x} + \text{ME}\right)
\]

Now let's perform these calculations step by step.

### Step 1: Calculate the sample mean (\(\bar{x}\))

The sales data is:
\[ 93, 117, 78, 130, 122, 117, 98, 102, 91, 101 \]

\[
\bar{x} = \frac{93 + 117 + 78 + 130 + 122 + 117 + 98 + 102 + 91 + 101}{10} = \frac{1049}{10} = 104.9
\]

### Step 2: Calculate the sample standard deviation (s)

First, compute the variance:

\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\]

Let's calculate each squared difference from the mean:

\[
(93 - 104.9)^2 = 138.81
\]
\[
(117 - 104.9)^2 = 145.21
\]
\[
(78 - 104.9)^2 = 717.61
\]
\[
(130 - 104.9)^2 = 625.21
\]
\[
(122 - 104.9)^2 = 289.21
\]
\[
(117 - 104.9)^2 = 145.21
\]
\[
(98 - 104.9)^2 = 47.61
\]
\[
(102 - 104.9)^2 = 8.41
\]
\[
(91 - 104.9)^2 = 194.41
\]
\[
(101 - 104.9)^2 = 15.21
\]

Sum of squared differences:

\[
\text{Sum of squared differences} = 138.81 + 145.21 + 717.61 + 625.21 + 289.21 + 145.21 + 47.61 + 8.41 + 194.41 + 15.21 = 2326.9
\]

Now, calculate the sample variance:

\[
s^2 = \frac{2326.9}{9} = 258.5444
\]

And the sample standard deviation:

\[
s = \sqrt{258.5444} \approx 16.08
\]

### Step 3: Determine the t-score

For a 90% confidence interval with 9 degrees of freedom (since \(n - 1 = 10 - 1 = 9\)), the t-score is approximately 1.833.

### Step 4: Calculate the margin of error (ME)

\[
\text{ME} = 1.833 \times \frac{16.08}{\sqrt{10}} \approx 1.833 \times 5.085 = 9.319
\]

### Step 5: Determine the confidence interval

\[
\text{CI} = (104.9 - 9.319, 104.9 + 9.319) = (95.58, 114.22)
\]

### Answer
The lower limit of the 90% confidence interval is **95.58**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How does the sample size affect the confidence interval?
2. What is the upper limit of the 90% confidence interval?
3. How would the confidence interval change if we used a 95% confidence level?
4. Can you explain why the t-score is used instead of the z-score in this scenario?
5. What happens to the confidence interval if the sample mean increases?
6. How does the variability (standard deviation) of the data influence the width of the confidence interval?
7. If we added more days to the dataset, how would it affect the confidence interval?
8. What is the interpretation of this confidence interval in the context of sales?

**Tip:** The confidence interval gives a range where the true population mean is likely to fall. It's a useful way to estimate the uncertainty around your sample mean.

Learn how to calculate the 90% confidence interval for the true mean of daily Eternal Roses sales using sample data. Understand the steps involving sample mean, sample standard deviation, t-score determination, margin of error calculation, and interpretation of confidence intervals.

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