t-Test

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This post covers t-Tests.

t-Distribution

  • Z - test works when we know $\mu$ and $\sigma$
  • Use Samples
    • How different a sample mean is from a population
    • How different two sample means are from each other
      • Two samples can be
        • Independent
        • Dependent
  • Estimate Population Standard Deviation using sample standard deviation with Bessel’s correction
    • Bessel’s correction is the use of $n − 1$ instead of $n$ in the formula for the sample variance and sample standard deviation, where $n$ is the number of observations in a sample.
    • This method corrects the bias in the estimation of the population variance.
    • It also partially corrects the bias in the estimation of the population standard deviation.
    • However, the correction often increases the mean squared error in these estimations.
    • This technique is named after Friedrich Bessel.
  • To find out how typical or atypical (unusual) a sample mean - find its location on the distribution of sample means i.e. sampling distribution
    • we can determine when we know population parameters, $\mu, \sigma$
    • $ std~errro= \frac{\sigma}{\sqrt{n}}$
    • $z = \frac{sample~mean - \mu}{std~error} = \frac{mean~difference}{std~error}$
    • Std for Samples = $S = \sqrt{\frac{\Sigma(X_i - \bar{X})^2}{n-1}}$
  • Standard Error depends on sample, we cannot use $\sigma$ if we have sample
  • Thus, we have a new distribution that is more prone to error - t-Distribution
    • more spread out and thicker in the tails than a normal distribution
      • Since large sample sizes gives skinnier sampling distribution
  • What happens as n increases?
    • The t-Distribution approaches to Normal Distribution
    • The t-Distribution gets Skinnier tails
    • $S \rightarrow \sigma$

Degree of Freedom - Sample Standard Deviation

  • We can pick a sample of size $n$ from population using $n$ degrees of freedom
  • Now to compute Standard Deviation, we need sample mean
  • $\bar{X} = \frac{X_1+X_2+X_3+…+X_n}{n}$
  • $ X_1+X_2+X_3+…+X_n = n . \bar{X} $
    • $n-1$ Degrees of Freedom
    • We may vary $n-1$ values to keep the sum of these values as $n\bar{X}$
    • $n-1$ is the effective sample size since only $n-1$ values are independent if we know the mean.
    • $S = \sqrt{\frac{\Sigma(X_i - \bar{X})^2}{n-1}}$
  • As degrees of freedom increases, the t-distribution better approxiamate the normal distribution

t-Table

  • https://naneja.github.io/python/t-table
  • Questions

      1. What’s the t-critical value for a one-tailed alpha level of 0.05 with 12 degrees of freedom.

        • df = 11 and p = 0.05
          • Ans t = 1.782

      2. What are t-critical values for 2-tailed test with $\alpha = 0.05$ and sample size 30

        • $df = 29,~ p = \pm 0.025$
        • Ans: $\pm 2.045$

      3. What are the limits for the right area of t-statistic when the sample size is 24 and the t-statistic is 2.45

        • $df=23, t=2.45$
    • $p = .02 ~\text{or}~ .01$

t-Statistic

$t = \frac{\bar{X}-\mu_0}{\frac{S}{\sqrt{n}}}$

  • The larger/smaller the value of $\bar{X}$, the stronger the evidence that $\mu > \mu_0$
  • The larger/smaller the value of $\bar{X}$, the stronger the evidence that $\mu < \mu_0$
  • The further the value of $\bar{X}$ from $\mu_0$ in either direction, the stronger/weaker the evidence that $\mu \ne \mu_0$

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One Sample t-Test

  • $t = \frac{\bar{X}-\mu_0}{\frac{S}{\sqrt{n}}}$

    \[H_0: \mu = \mu_0 \\\begin{align*} H_A &: \mu < \mu_0 \\ &: \mu > \mu_0 \\ &: \mu \ne \mu_0 \end{align*}\]

  • $\alpha$ Levels (column levels of t-table)

  • What will increase the t-Statistic
    • Large difference between $\bar{X}$ and $\mu_0$
    • Larger $n$
    • Larger $S$
    • Large Standard Error
  • Larger t-Statistic
    • => Lower probability of obtaining t-Statistic
    • => Larger $\bar{X} - \mu_0$

P-Value

  • Compute t-statistic

    • $t = \frac{\bar{X}-\mu_0}{\frac{S}{\sqrt{n}}}$

  • One-tailed Test

    • p-value is the probability
      • above the t-Statistic if it’s positive, or
      • below the t-Statistic if it’s negative

  • Two-tailed Test

    • p-value is the probability of the sum of both
      • above the t-Statistic and
      • below the t-Statistic

  • Reject the Null when the p-value is less than the $\alpha$ level

Example - Finches Beek Width

  • Average known Beak Width = 6.07 mm
  • $H_0: \mu = 6.07$
  • $H_A: \mu \ne 6.07$
    • Sample Size = 500
    • Degrees of Freedom = 499
  • Compute the sample mean and std dev from the sample dataset
    • $\bar{X} = 6.470$
    • $S = \sqrt{\frac{\Sigma(X_i - \bar{X})^2}{n-1}} = 0.396$
  • t-Statistic
    • $ t = \frac{6.47 - 6.07}{0.396/\sqrt{500}} = \frac{0.4}{0.0179} = 22.346$
  • Table
    • $df = 499,~ t = 22.346$, https://naneja.github.io/python/t-table
    • $p = 0$
    • Reject null
      • probability of getting this t-value is very very small
      • probability of getting the sample with beek width 6.47 from the population with mean 6.07 is very very small

Example

  • Sample = [5, 19, 11, 23, 12, 7, 3, 21]

  • Is this sample mean significantly different from 10 at an alpha level of 0.05?

    • Different => two-tailed t-test
    • $n=8,~ \bar{X} = 12.625, S = 7.6$
    • $t = \frac{\bar{X} - 10}{\frac{S}{\sqrt{n}}} = \frac{12.625 - 10}{\frac{7.6}{\sqrt{8}}} = 0.977$
    • $df=7, t=0.977$
    • https://naneja.github.io/python/t-table
    • Two Tail Test
      • $ p = 0.18 + 0.18 = 0.36 > 0.05 $
    • Fail to Reject
      • Thus, $H_0: \mu = 10$

Example

  • Mean Rent = 1830 for all apartments

  • Company A wants to know if the rent they are charging is significantly different at $\alpha = 0.05$

    • Sample:
      • $n=25,~ \bar{X}=1700,~ S=200$

  • Hypothesis

    • $H_0: \mu = 1830$ and
    • $H_A: \mu \ne 1830$
    • What are t-critical values
      • $t = \frac{\bar{X} - 1830}{\frac{S}{\sqrt{n}}} = \frac{1700 - 1830}{\frac{200}{\sqrt{25}}} = -3.250$
      • $df = 24, t = -3.25 $
      • Two Tail Test
      • https://naneja.github.io/python/t-table
      • $p = 0.0025 + 0.0025 = 0.005 < 0.05$
    • Reject the null in favor of $H_A: \mu \ne 1830$

  • What is the Confidence Interval for the population for Company A?

    • 95% Confidence Interval

    • For 95% CI, we need t-critical value at p=0.025 at df = 24

      • $df=24, ~p=0.025 \implies t_{critical} = 2.064 $
      • $\pm ~\text{t_critical} * \text{std_error} = t * \frac{S}{\sqrt{n}} $
      • $2.064 * \frac{200}{\sqrt{25}} = 82.56$
      • CI = $(1700 - 82.56, 1700 + 82.56) ~=~ (1617.44, 1782.56) $

    • Margin of Error = 82.56

  • If n = 100

    • What are t-critical values

      • $t = \frac{\bar{X} - 1830}{\frac{S}{\sqrt{n}}} = \frac{1700 - 1830}{\frac{200}{\sqrt{100}}} = -6.50$
      • $df = 24, t = 6.5 $
        • https://naneja.github.io/python/t-table
        • $p = 0.00 + 0.00 = 0.00 < 0.05$

    • Reject the null in favor of $H_A: \mu \ne 1830$

    • What is the Confidence Interval for the population for Company A? - 95% Confidence Interval

      • $df = 99, p = 0.025 \implies t_{critical} = $ 1.984
      • Margin of Error = $\pm ~\text{t_critical} * \text{std_error} = t_{critical} * \frac{S}{\sqrt{n}} $
      • ME = $1.984 * \frac{200}{\sqrt{100}} = 39.68$
      • Increase of sample size will reduce Margin of error
      • CI = $(1700 - 39.68, 1700 + 39.68) ~=~ (1660.32, 1739.68) $

Cohen’s d

  • Standardized mean difference that measures the distance between means in standardized units
  • $Cohen’s~d = \frac{\bar{X}-\mu}{S}$
  • In above Rent Example
    • $\mu = 1830$
    • Sample: $n = 25, \bar{X} = 1700, S = 200$
    • $d = \frac{\bar{X}-\mu}{S} = \frac{1700 - 1830}{200} = -0.65$

Dependent Samples

  • Same subject takes the test twice

  • Within subject designs

    • each subject is assigned two conditions in random order

      • in control but get treatment

      • two kinds of treatment

    • Every subject is given a Pre-Test and a Post-Test

    • Growth over time (Longitudinal Study)
      • Each subject at different points of time
    xiyiDi = xi - yi
    x1y1D1 = x1-y1
    x2y2D2 = x2-y2
    x3y3D3 = x3-y3

​

Example - Keyboards

  • Errors in two design of keyboards (QWERTY and Alphabetical)
    • https://naneja.github.io/datasets
      • file = Keyboards.csv
  • Mean Error
    • n = 25
    • Querty Keyboard = 5.08 and
    • Alphabetical Keyboard = 7.8

  • Are these differences significant?

    • $n = 25$

    • $H_0: \mu_Q = \mu_A ~and~ H_A: \mu_Q \ne \mu_A$

      • Also can say $\mu_Q - \mu_A = 0$

    • What is Point Estimate for $\mu_Q - \mu_A$

      • $\mu_Q - \mu_A = 5.08 - 7.8 = -2.72$

    • What is S

      • d = QWERTYerrors - Alphabeticalerrors
      • std_dev = $S = s_d = \sqrt{\frac{\Sigma (d-\bar{d})^2}{N-1}}$
      • 3.69
    • What is t-Statistic when S = 3.69
      • $t = \frac{\mu_Q - \mu_A}{\frac{S}{\sqrt{n}}} = \frac{5.08 - 7.8}{\frac{3.69}{\sqrt{25}}} = -3.69$
    • What is p-value
      • Two Tail Test
        • $df=24, t = -3.69$
        • https://naneja.github.io/python/t-table
      • $ p = 0.0005 + 0.0005 = 0.001 < 0.05 $

    • Reject the Null or Fail to reject Null

      • Reject the Null
    • Significant Less Error and we may say causal effect due to keyboard design
    • 95% Confidence Interval
      • What are t-Critical Values for $\alpha=0.05$
        • $ df = 24, p = 0.025$
        • $t_{critical} = \pm 2.064$
      • point estimate$ = \mu_Q - \mu_A = 5.08 - 7.8 = -2.72$
      • $CI = \text{point estimate} \pm t_{critical} * \text{std error}$
      • $CI = -2.72 - 2.064 * \frac{3.69}{\sqrt{25}}, ~ -2.72 + 2.064 * \frac{3.69}{\sqrt{25}}$
      • -4.24, -1.20
      • Users will make fewer errors in the range of 4 to 1 on querty keyboard than alpha errors

Advantages and Disadvantages- Dependent Samples

    • Within-Subject design
      • Two Conditions
      • Longitudinal
      • Pre-Test, Post-Test
    • Advantages
      • Controls for individual differences
      • Use Fewer Subjects
      • Cost-Effective
      • Less Time-consuming
      • Less Expensive
    • Disadvantages
      • Carry-over Effects
      • Second measurement can be affected by first treatment
      • Order may influence results

Independent Samples

  • Between-Subject Designs

    • Experimental

    • Observational

    \[H_0: \mu_1 - \mu_2 = 0 \\\begin{align*} H_A &: \mu_1 - \mu_2 > 0 \\ &: \mu_1 - \mu_2 < 0 \\ &: \mu_1 - \mu_2 \ne 0 \end{align*}\]

  • $ t = \frac{\bar{X_1}-\bar{X_2}}{standard~error} $

  • Standard Deviation = $\sqrt{S_1^2 + S_2^2}$

    • We can’t take differences since these are not the same subjects (independent samples)

  • Standard Error $ = \frac{S}{\sqrt{n}} = \frac{\sqrt{S_1^2 + S_2^2}}{\sqrt{n}} = \sqrt{\frac{S_1^2 + S_2^2}{n}} = \sqrt{\frac{S1^2}{n} + \frac{S2^2}{n}} = \sqrt{\frac{S1^2}{n_1} + \frac{S2^2}{n_2}}$

  • Degrees of Freedom $ df= (n_1-1) + (n_2-1) = n_1 + n_2 -2$

  • $ t = \frac{(\bar{X_1}-\bar{X_2})}{SE}$

Example - Food Prices

  • https://naneja.github.io/datasets
    • file = FoodPrices.csv
  • Hypothesis
    • $H_0 : \mu_1 = \mu_2$
    • $H_A : \mu_1 \ne \mu_2$
  • $n1 = 18,~ n2 = 14$
  • $df = n1 + n2 - 2 = 30$
  • Sample Mean and Std Dev
    • $\bar{X_1} = 8.94,~ \bar{X_2} = 11.14$
    • $S_1 = 2.65,~ S_2 = 2.18$
  • $S = \sqrt{\frac{S1^2}{n_1} + \frac{S2^2}{n_2}} = \sqrt{\frac{2.65^2}{18} + \frac{2.18^2}{14}} =0.85$
  • $t = \frac{\bar{X_1} - \bar{X_2}}{S} = \frac{8.94 - 11.14}{0.85} =2.58$ (Ignore negative sign)
  • $df = 30,~ t = 2.58$
  • $p = 0.01 + 0.01 = 0.02 < 0.05$
  • Reject the Null
    • Prices are significantly different for both areas

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