Interval estimation is a range of values used to estimate a population parameter. Unlike point estimation, interval estimation provides a range that is likely to contain the true population parameter with a specified level of confidence.

• Point Estimate: A single value estimate of a population parameter (e.g., sample mean).
• Confidence Interval (CI): A range of values that is likely to contain the population parameter with a certain level of confidence.
• Confidence Level: The proportion of times that the confidence interval would contain the true population parameter if repeated sampling were done.

The confidence interval for the population mean (when the population standard deviation is unknown) is given by:

\[ CI = \bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} \]

Where: - \(\bar{x}\) is the sample mean. - \(t_{\alpha/2, n-1}\) is the critical value from the t-distribution. - \(s\) is the sample standard deviation. - \(n\) is the sample size.

For a one-tailed test, the P-value is calculated as:

\[ P = P(t > t_{observed}) \]

Where \(t_{observed}\) is the observed value of the test statistic.

Let’s compute a 95% confidence interval for a sample mean.

## 
##  One Sample t-test
## 
## data:  data
## t = 36.027, df = 29, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  60.86573 68.19219
## sample estimates:
## mean of x 
##  64.52896

We use ggplot2 to visualize the distribution of the sample data.

This plot shows the confidence interval for the sample mean.

library(ggplot2)

ggplot(data = data.frame(x=data), aes(x=x)) +

geom_histogram(aes(y=after_stat(density)), binwidth=2, fill=“brown”, alpha=0.5) +

geom_density(color=“grey”, linewidth=1.5) + # Changed size to linewidth

ggtitle(“Distribution of Sample Data”)

Interval estimation provides a range of values that likely contains the true population parameter, giving a measure of reliability to the estimate. Confidence intervals are widely used in research and data analysis to quantify uncertainty.