MTH-161D | Spring 2025 | University of Portland
April 2, 2025
The \(t\)-distribution
The larger the degrees of freedom the more closely the \(t\)-distribution resembles the standard normal distribution.
Confidence Intervals for One Mean
\[ \begin{aligned} \bar{x} \ &\pm\ t^*_{df} \times \frac{s}{\sqrt{n}} \end{aligned} \]
\[ \begin{aligned} \bar{x} & \longrightarrow \text{sample mean (point estimate)} \\ s & \longrightarrow \text{sample standard deviation} \\ n & \longrightarrow \text{sample size} \\ t^*_{df} & \longrightarrow \text{critical value (t-distribution with degrees of freedom } df \text{)} \end{aligned} \]
The problem shown below was taken and slightly modified from your textbook OpenIntro: Introduction to Modern Statistics Section 20.6. Consider the research study described below.
Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions manufactured in 2021. Do these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? US DOE EPA 2021
We will compute the 95% confidence interval for the true difference in means \(\mu_{automatic} - \mu_{manual}\).
CITY | Mean | SD | n |
---|---|---|---|
Automatic | 17.44 | 3.44 | 25 |
Manual | 22.68 | 4.58 | 25 |
Here, we see two outliers in the manual group. However, both groups shows decent distributions with balanced outliers where - in this case with 25 samples each - we can “ignore” the outliers and assume normality of the sampling distribution of the means.
Two-sample t-intervals
\[ \begin{aligned} \text{point estimate} \ &\pm\ t^*_{df} \times SE \\ \bar{x}_1 - \bar{x}_2 \ &\pm\ t^*_{df} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \end{aligned} \]
Standard Error \[ \begin{aligned} SE & = \sqrt{\frac{s_{automatic}^2}{n_{automatic}} + \frac{s_{manual}^2}{n_{manual}}} \\ & = \sqrt{\frac{3.44^2}{25} + \frac{4.58^2}{25}} \\ SE & = 1.1456 \end{aligned} \]
Degrees of freedom is \(df = 24\).
For a 95% confidence level, we find the the critical \(t^*_{df}\) where the upper tail is equal to 2.5%: \(t^*_{24} = 2.0639.\) The area below \(t^*_{24} = -2.0639\) will also be equal to 2.5%.
Using R to find the critical \(t^*\)
cl <- 0.95 # confidence level
lt <- (1-cl)/2 # lower tail probability
df <- 24 # degrees of freedom
qt(lt, df) # t-star
## [1] -2.063899
Therefore, we are 95% confident that the true difference in mean fuel efficiency (miles/gallon) between automatic and manual cars is between 2.9356 and 7.664 in absolute value.
Note that the values are originally negative because how the order of difference terms are computed, meaning a negative difference indicate that there is more efficiency in cars with manual transmission than automatic transmission.
One-sample t-interval
\[df = n - 1\]
Two-sample t-interval
If the population variance is unknown, use the sample variance \(s^2\). If the population variance is known, use population variance \(\sigma^2\). Most real-world problems involve sample variances, especially for statistical inference.
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