Hypothesis test for a mean

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Given Problem : Two-Tailed Test

Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

This approach consists of four steps:

1. State the hypotheses

The first step is to state the null hypothesis and an alternative hypothesis.

\[Null\ hypothesis(H_0): \mu = 23\] \[Alternative\ hypothesis(H_1): \mu \neq 23\]

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the sample mean is too big or if it is too small..

2. Formulate an analysis plan

For this analysis, the significance level is 95%. The test method is a one-sample t-test.

3. Analyze sample data.

Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

\[SE = \frac{s}{\sqrt{n}} = \frac{6.0269481}{\sqrt{32}} = 1.07\] \[DF=n-1=32-1=31\]

\[t = (\bar{x} - \mu) / SE = (20.090625 - 23)/1.07 = -2.731\]

where s is the standard deviation of the sample, \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized population mean, and n is the sample size.

We can visualize the confidence interval of mean.


Since we have a two-tailed test, the P-value is the probability that the t statistic having 31 degrees of freedom is less than -2.73 or greater than 2.73.

We use the t Distribution curve to find p value.


\[pt(-2.731,31) =0.01 \]

4. Interpret results.

Since the P-value (0.01) is less than the significance level (0.05), we can reject the null hypothesis.

Result of meanCI()

call: meanCI.data.frame(x = mtcars, mpg, mu = 23) 
method: One sample t-test 
alternative hypothesis:
   true mean  is not equal to  23 

# 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 -2.7307 0.01033


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