4. Inference for numerical data

7. Inference for numerical data ¹

T distribution ²

• T distribution is useful to plot the distribution of the sample mean when the population standard deviation is unknown.
• It is a bell-shaped distribution that is symmetric around 0.
• It is similar to the normal distribution but has heavier tails, and lower peak.
• Observations are more likely to fall in the tails of the t-distribution than the normal distribution.
• Observations are more likely to fall beyond 2 standard deviations from the mean in the t-distribution than the normal distribution.
• Confidence intervals are wider, aka, more conservative when using the t-distribution than the normal distribution.
• Thick tails means more error for mitigating the uncertainty and the less reliable the estimate for standard error.
• T distribution has a parameter called degrees of freedom which determines the thickness of the tails.
• As the degrees of freedom increase, the t-distribution approaches the normal distribution.
• calculate the t statistic using the formula: $T=\frac{obs-null}{SE}$.
• P-value is the probability of observing a test statistic as extreme as the one observed, assuming the null hypothesis is true.

Inference for a mean ³

• Mean of the population is within the confidence interval.

$$\begin{array}{rl}pointestimate& \pm marginoferror\\ \overline{x}& \pm t_{df}^{\ast }S{E}_{\overline{x}}\\ \overline{x}& \pm t_{df}^{\ast }\frac{s}{\sqrt{n}}\\ \overline{x}& \pm t_{n-1}^{\ast }\frac{s}{\sqrt{n}}\end{array}$$

• Where:
  • s is the sample standard deviation.
  • n is the sample size.
  • t* is the t-score.
  • df is the degrees of freedom.
  • SE is the standard error.
  • x is the sample mean.
• To find the t* score:
  • Calculate the degrees of freedom: $df=n-1$.
  • Use the t-distribution table to find the t* score:
    • Find the row that corresponds to the degrees of freedom.
    • Find the column that corresponds to the confidence level.
    • The value at the intersection is the t* score.
  • Or use the qt() function in R to find the t* score:
    • qt((1 - confidence)/2, df = n - 1)
    • for 95% confidence level, qt(0.025, df = n - 1)
• To compute the p-value:
  • Calculate the degrees of freedom: $df=n-1$.
  • Calculate the t statistic using the formula: $t=\frac{obs-null}{SE}$.
  • Use the pt() function in R to find the p-value:
    • pt(t, df, lower.tail = FALSE) * 2
    • for a two-tailed test, multiply by 2.
  • Using the t-distribution table:
    • Find the row that corresponds to the degrees of freedom.
    • Find the column that corresponds to the t statistic.
    • The value at the intersection is the p-value.

Inference for paired data ⁴

• Paired data is when two observations are linked in some way, aka, not independent.
• The difference between the two observations is calculated, and it is used to perform inference.
• The difference is the new data set that is used to calculate the mean and standard deviation.
• If the average difference is 0, then the null hypothesis is true; and there is no difference between the two observations sets.
• H0: ${\mu }_{diff}=0$. Ha: ${\mu }_{diff}\ne 0$.

Difference of two independent means ⁵

$$\begin{array}{rl}pointestimate& \pm marginoferror\\ ({\overline{x}}_1-{\overline{x}}_2)& \pm t_{df}^{\ast }S{E}_{{\overline{x}}_1-{\overline{x}}_2}\\ ({\overline{x}}_1-{\overline{x}}_2)& \pm t_{df}^{\ast }\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ ({\overline{x}}_1-{\overline{x}}_2)& \pm t_{(min(n_1-1,n_2-2))}^{\ast }\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\end{array}$$

So:

$$\begin{gathered} S{E}_{{\overline{x}}_1-{\overline{x}}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \\ df=min(n_1-1,n_2-2) \end{gathered}$$

• Conditions:
  • Independence:
    • Within groups: sampled observations are independent.
      • Random sample/assignment.
      • 10% condition: both sample sizes are less than 10% of the population.
    • Between groups: the two groups are independent.
      • Groups are not paired.
  • Sample size/skew:
    • The more skewed the data, the larger the sample size required.

References

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1. Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2019). Openintro statistics - Fourth edition. Open Textbook Library. https://www.biostat.jhsph.edu/~iruczins/teaching/books/2019.openintro.statistics.pdf Chapter 7 - Inference for numerical data. Section 7.1 - One Sample means with t-distribution from page 251 to page 261 Section 7.2 - Paired data from page 262 to page 266
   Section 7.3 - Difference of two means from page 267 to page 277 ↩

2. Çetinkaya-Rundel, M. (2018a, February 20). 5 1A t distribution [Video]. YouTube. https://youtu.be/uVEj2uBJfq0 ↩

3. Çetinkaya-Rundel, M. (2018b, February 20). 5 1B Inference for a mean [Video]. YouTube. https://youtu.be/RYVIGj1l4xs ↩

4. Çetinkaya-Rundel, M. (2018c, February 20). 5 2 Inference for paired data [Video]. YouTube. https://youtu.be/K0QZ9_4w0HU ↩

5. Çetinkaya-Rundel, M. (2018d, February 20). 5 3 Difference of two independent means [Video]. YouTube. https://youtu.be/emZ24asR2F4 ↩