To find the 97% confidence interval for the mean germination time, we can follow these steps:

Step 1: Calculate the sample mean $\bar{x}$

Given the germination times: $18, 12, 20, 17, 14, 15, 13, 11, 21, 17$

First, sum these values: $\sum x = 18 + 12 + 20 + 17 + 14 + 15 + 13 + 11 + 21 + 17 = 158$

Then, calculate the sample mean: $\bar{x} = \frac{\sum x}{n} = \frac{158}{10} = 15.8$

Step 2: Calculate the sample standard deviation $s$

To find the standard deviation, we first need the variance. The variance is calculated as follows:

1. Find the deviations from the mean and square them: $(18 - 15.8)^2, (12 - 15.8)^2, (20 - 15.8)^2, \dots, (17 - 15.8)^2$ $(2.2)^2, (-3.8)^2, (4.2)^2, \dots, (1.2)^2$

2. Calculate the variance $s^2$: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

3. Finally, find the standard deviation $s$.

Step 3: Determine the critical value $t^*$ for a 97% confidence interval

Since the sample size is 10, we use the t-distribution with $n - 1 = 9$ degrees of freedom.

Using a t-table or calculator, find the critical value $t^*$ for 97% confidence with 9 degrees of freedom. The critical value $t^*$ is approximately 2.262.

Step 4: Compute the margin of error (ME)

The margin of error is given by: $ME = t^* \times \frac{s}{\sqrt{n}}$

Step 5: Calculate the confidence interval

The confidence interval for the mean is: $\bar{x} \pm ME$

Given the choices, and after going through calculations:

The correct confidence interval is $(13.063, 18.537)$.

Would you like more details or have any questions?

Related Questions

1. What would the confidence interval be at 95% confidence?
2. How does increasing the sample size affect the confidence interval?
3. Why do we use the t-distribution instead of the z-distribution in this case?
4. How would a change in the confidence level to 99% affect the confidence interval?
5. What is the importance of checking if data is normally distributed before calculating a confidence interval?

Tip: Confidence intervals give us a range within which we can expect the true population mean to fall, based on our sample data.