Recorded here are the germination times in days for 20 randomly chosen seeds of a new type of bean 21,22, 15 15, 17, 17, 11, 13, 18,21, 12, 14, 12, 13, 10, 19, 22, 13, 18, 15. Assuming that germination times are normally distributed find a 90% confidence interval for the mean germination time for all beans of this type. give the lower limit and the upper limit of the 90% confidence interval. Carry your intermediate computations to at least three decimal places. round your answers to one decimal place.

To find the 90% confidence interval for the mean germination time of all beans of this type, we'll follow these steps:

### Step 1: Identify the given values
- We have a sample of germination times: 21, 22, 15, 15, 17, 17, 11, 13, 18, 21, 12, 14, 12, 13, 10, 19, 22, 13, 18, 15.
- Sample size (\(n\)) = 20 (since there are 20 values).
- Since the population standard deviation is unknown and the sample size is small (\(n < 30\)), we'll use the t-distribution.

### Step 2: Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\))
The sample mean (\(\bar{x}\)) and the sample standard deviation (\(s\)) are calculated using the formulas:

\[
\bar{x} = \frac{\sum x_i}{n}
\]

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

### Step 3: Find the t-score for a 90% confidence level with \(n-1\) degrees of freedom
For a 90% confidence level and \(n-1 = 19\) degrees of freedom, we'll find the t-score (\(t_{\alpha/2}\)).

### Step 4: Calculate the standard error (SE)
The standard error (SE) is given by:

\[
\text{SE} = \frac{s}{\sqrt{n}}
\]

### Step 5: Calculate the margin of error (ME)
The margin of error (ME) is given by:

\[
\text{ME} = t_{\alpha/2} \cdot \text{SE}
\]

### Step 6: Calculate the confidence interval
The confidence interval is:

\[
\text{Confidence Interval} = \bar{x} \pm \text{ME}
\]

Let's calculate these values.The 90% confidence interval for the mean germination time of all beans of this type is approximately:

- **Lower limit:** 14.4 days
- **Upper limit:** 17.4 days

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. What does it mean if a value falls outside of the confidence interval?
2. How is the t-distribution different from the normal distribution?
3. What would happen to the confidence interval if we had a larger sample size?
4. Why is the sample standard deviation used instead of the population standard deviation?
5. How does the level of confidence affect the width of the confidence interval?
6. What is the impact of sample variability on the confidence interval?
7. How do you interpret the results of a confidence interval in practical terms?
8. What are the assumptions needed for a valid confidence interval using the t-distribution?

**Tip:** When dealing with small sample sizes, it's important to use the t-distribution rather than the normal distribution to account for the increased uncertainty.

Learn how to calculate a 90% confidence interval for the mean germination time of a new type of bean using statistical methods. This problem involves using the t-distribution due to a small sample size and unknown population standard deviation. Understand the steps involved in computing sample mean, sample standard deviation, standard error, margin of error, and the final confidence interval.

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