Questions: 1. You have collected data on the rainfall (inches) in Columbia for 20 randomly selected days. Read the rain data into R in order to answer the following questions: (a) Construct and interpret a 95% confidence interval for the average amount of rainfall per day in Columbia. CI- 0.3250 .739 which means that we are 95% confident that the true average rainfall per day in Columbia lies within this range (b) Based on your confidence interval, do we have evidence that it rains less than half an inch per day in Columbia on average? No we do not have evidence that is rains less than half an inch per day in Columbia on average because the CI(0.325 to 0.739) is above 0.5 inches (c) Are the assumptions of this statistical procedure satisfied? What does that mean?

Solution Steps

The mean rainfall ($\mu$) for the 20 randomly selected days is calculated as follows:

$$\mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{10.8}{20} = 0.54 \text{ inches}$$

The sample size ($n$) is:

$$n = 20$$

The sample standard deviation ($s$) is calculated to be:

$$s = 0.17 \text{ inches}$$

Using the formula for the confidence interval for the mean of a single population with unknown variance and small sample size, we have:

$$\bar{x} \pm t \frac{s}{\sqrt{n}} = 0.54 \pm 2.093 \cdot \frac{0.17}{\sqrt{20}}$$

Calculating the confidence interval:

$$= 0.54 \pm 0.08 \implies (0.46, 0.62)$$

Thus, the 95% confidence interval is:

$$\text{Confidence Interval: } (0.46, 0.62)$$

Based on the confidence interval, we check if there is evidence that it rains less than half an inch per day in Columbia on average. Since the entire confidence interval $(0.46, 0.62)$ is above $0.5$ inches, we conclude:

$$\text{No, we do not have evidence that it rains less than half an inch per day in Columbia on average.}$$

The assumptions of the statistical procedure are satisfied if the sample is random and the population is normally distributed or if the sample size is large enough (by the Central Limit Theorem). Therefore, we conclude:

$$\text{Yes, the assumptions of the statistical procedure are satisfied.}$$

Final Answer