# Confidence interval for a mean

This document is prepared automatically using the following R command.

## Problem

## Confidence interval of mean

The approach that we used to solve this problem is valid when the following conditions are met.

- The sampling method must be simple random sampling.

- The sampling distribution should be approximately normally distributed.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval of mean.

### Raw data

The first 10 rows of the provided data is as follows.

    # A tibble: 10 × 1
         mpg
       <dbl>
     1  21
     2  21
     3  22.8
     4  21.4
     5  18.7
     6  18.1
     7  14.3
     8  24.4
     9  22.8
    10  19.2

### Sample statistics

The sample size is 32, the sample mean is 20.09 and the standard error of sample is 6.03. The confidence level is 95 %.

### Find the margin of error

Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean; instead, we compute the standard error (SE). Because the sample size is much smaller than the population size, we can use the “approximate” formula for the standard error.

\[ SE= \frac{s}{\sqrt{n}}\] where s is the standard deviation of the sample, n is the sample size.

\[SE=\frac{6.03}{\sqrt{32}}=1.07\] Find the critical probability(p*):

\[p*=1-\alpha/2=1-0.05/2=0.975\]

The degree of freedom(df) is: \[df=n-1=32-1=31\]

The critical value is the t statistic having 31 degrees of freedom and a cumulative probability equal to 0.975. From the t Distribution table, we find that the critical value is 2.04.

\[qt(p,df)=qt(0.975,31)=2.04\]

The graph shows the \(\alpha\) values are the tail areas of the distribution.

Compute margin of error(ME):

\[ME=critical\ value \times SE\] \[ME=2.04 \times 1.065=2.173\]

Specify the confidence interval. The range of the confidence interval is defined by the sample statistic \(\pm\)margin of error. And the uncertainty is denoted by the confidence level.

### Confidence interval of the mean

Therefore, the 95% confidence interval is 17.92 to 22.26. That is, we are 95% confident that the true population mean is in the range 17.92 to 22.26.

### Plot

You can visualize the mean difference:

### Result of meanCI()

    call: meanCI.data.frame(x = mtcars, mpg)
    method: One sample t-test
    alternative hypothesis:
       true mean  is not equal to  0

    Results
    # A tibble: 1 × 7
      m        se     DF    lower    upper    t      p
      <chr>    <chr>  <chr> <chr>    <chr>    <chr>  <chr>
    1 20.09062 1.0654 31    17.91768 22.26357 18.857 < 2.2e-16

### Reference

The contents of this document are modified from StatTrek.com. Berman H.B., “AP Statistics Tutorial”, [online] Available at: https://stattrek.com/estimation/confidence-interval-mean.aspx?tutorial=AP URL[Accessed Data: 1/23/2022].