# t-Tests

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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 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

• Questions

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

• Ans 1.782

• p = 0.05
df = 12

# 1-p for right-tailed test
value = round(stats.t.ppf(1-p, df), 3)
print(value) # 1.782

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

• Ans: $\pm 2.045$

• p = 0.025
sample_size = 30
df = sample_size - 1

# p for left-tailed test
value = round(stats.t.ppf(p, df), 3)
print(value) # -2.045

# 1-p for right-tailed test
value = round(stats.t.ppf(1-p, df), 3)
print(value) # 2.045

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

• .02 and .01

• value = 2.45
sample_size = 24
df = sample_size - 1

p = round(1 - stats.t.cdf(value, df), 3)
print(p) # 0.011


# 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 $\mu \ne \mu_0$

# 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 $\bar{X} - \mu_0$

# 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 sample mean and std dev from the sample dataset
• $\bar{X} = 6.470$
• $S = 0.396$
• t-Statistic
• $t = \frac{6.47 - 6.07}{0.396/\sqrt{500}} = \frac{}{0.0179} = 22.346$
• Reject Null or Fail to reject Null
• Reject null since t-value is very large
• 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
• p-value
• probability of getting a t-statistic

# 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

• 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

• t = 0.977

• from scipy import stats
def sample_std(data):
xbar = np.mean(data)
std = [(d - xbar)**2 for d in data]

df = len(data)-1
std = np.sqrt(sum(std)/df)

return std

data = [5, 19, 11, 23, 12, 7, 3, 21]
xbar = np.mean(data)
print(xbar) # 12.625

n = len(data)
df = n - 1 # 7

S = sample_std(data)
print(S) # 7.6

t = (xbar - 10)/(S/np.sqrt(n))
print(f't={t:.3f}') # 0.977

• Since two-tailed test then
• $p = p(t<-0.9777) + p(t>0.9777)$
• From the table
• df = 7 and t = 0.977
• Left p = 0.18 [between 0.20 and 0.15]
• Similarly, Right p = 0.18 [between 0.20 and 0.15]
• since symmetrical
• $p = 0.36 ~(0.30 < p < .40)$
• https://www.socscistatistics.com/pvalues/tdistribution.aspx

• p = round(1 - stats.t.cdf(t, df), 3)
print(2*p) # 0.36

• $p$ is not statisticaly significant since $p=0.36 > \alpha = 0.05$, so we fail to reject the Null

• 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: $n=25,~ \bar{X}=1700,~ S=200$
• $H_0: \mu = 1830$ and $H_A: \mu \ne 1830$

• What are t-critical values

• t-Critical = $\pm 2.064$

  alpha = 0.05

sample_size = 25
df = sample_size-1

# Two tailed test, so alpha/2
t_critical = stats.t.ppf(alpha/2, df)

print(f't_critical = {t_critical:.3f} and {-t_critical:.3f}') # -2.064

• What is the t-statistic value

• $t = -3.25$

• $S = \sqrt{\frac{\Sigma(X_i - \bar{X})^2}{n-1}} = 200$

• $t = \frac{xbar - mu}{S/np.sqrt(n)}$

  mu = 1830
sample_size = 25
xbar = 1700
S = 200

t = (xbar - mu)/(S/np.sqrt(sample_size))
print(f't={t:.3f}') # -3.250

• t is in critical region so reject the null in favor of $H_A: \mu \ne 1830$

• Rental company charges significantly less that population 1830

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

• 95% Confidence Interval = (1617.44, 1782.56)

• $\pm ~\text{t_critical} * \text{std_error}$
• Margin of Error = 82.56

  std_error = S/np.sqrt(sample_size)

CI95_lb = xbar - abs(t_critical) * std_error
CI95_ub = xbar + abs(t_critical) * std_error

print(f'95% CI = {CI95_lb:.2f}, {CI95_ub:.2f}') # 1617.44, 1782.56

margin_of_error = abs(t_critical) * std_error
print(f'margin of error = {margin_of_error:.2f}') # 82.56

• If n = 100 then t_critical=-1.984 and Margin of Error = 39.68

• Increase of sample size will reduce Margin of error

  alpha = 0.05

sample_size = 100
df = sample_size-1

# Two tailed test, so alpha/2
t_critical = stats.t.ppf(alpha/2, df)

print(f't_critical = {t_critical:.3f} and {-t_critical:.3f}')
# -1.984 and 1.98

mu = 1830
xbar = 1700
S = 200

t = (xbar - mu)/(S/np.sqrt(sample_size))
print(f't={t:.3f}') # -6.500

std_error = S/np.sqrt(sample_size)

CI95_lb = xbar - abs(t_critical) * std_error
CI95_ub = xbar + abs(t_critical) * std_error

print(f'95% CI = {CI95_lb:.2f}, {CI95_ub:.2f}') # 1660.32, 1739.68

margin_of_error = abs(t_critical) * std_error
print(f'margin of error = {margin_of_error:.2f}') # 39.68


# Cohen’s d

• Standardised mean difference that measures the distance between means in standardised units

• $Cohen’s~d = \frac{\bar{X}-\mu}{S}$

• mu = 1830

n = 25
xbar = 1700
S = 200
alpha = 0.05

d = (xbar - mu)/S
print(f'd={d:.3f}') # -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)

• Mean Error on Querty Keyboard = 5.08 and Alphabetical Keyboard = 7.98

• import numpy as np

# https://naneja.github.io/datasets
file = './data/Keyboards.csv'

xbar_q = df.QWERTYerrors.mean()
xbar_a = df.Alphabeticalerrors.mean()
print(xbar_q, xbar_a) # 5.08 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 $\mu_Q - \mu_A$

• -2.72

• point_estimate = xbar_q - xbar_a
print(f'point_estimate={point_estimate:.3f}') # -2.720

• What is S

• 3.69

• S = df['d'].std(ddof=1) # 3.69
# delta degrees of freedom 1 for sample

df['d'] = (df.QWERTYerrors - df.Alphabeticalerrors)

# Compute S for d
m = df.d.mean()
df['S'] = (df.d - m)**2
S = np.sqrt(df.S.sum() / (df.shape[0] - 1))
print(f'S={S:.2f}')

• What is t-Statistic when S = 3.69

• t = -3.69

• S = 3.69
t = point_estimate / (S/np.sqrt(df.shape[0]))
print(f't={t:.2f}')

• What are t-Critical Values for $\alpha=0.05$

• $\pm 2.064$

• from scipy import stats
alpha = 0.05
t_critical = stats.t.ppf(alpha/2, df.shape[0]-1)
print(f't_critical= pm {abs(t_critical):.3f}') # -2.064

• 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

• -4.24, -1.20

• std_error = S/np.sqrt(df.shape[0])

CI95_lb = point_estimate - abs(t_critical) * std_error
CI95_ub = point_estimate + abs(t_critical) * std_error
print(f'95% CI = {CI95_lb:.2f}, {CI95_ub:.2f}') # -4.24, -1.20

margin_of_error = abs(t_critical) * S/np.sqrt(df.shape[0])
print(f'margin of error = {margin_of_error:.2f}') # 1.52

• Users will make fewer errors in the range of 4 to 1 on querty keyboard than alpha errors

• Within-Subject design
• Two Conditions
• Longitudinal
• Pre-Test, Post-Test
• Controls for individual differences
• Use Fewer Subjects
• Cost-Effective
• Less Time-consuming
• Less Expensive
• 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*}
• $t = \frac{\bar{X_1}-\bar{X_2}}{standard~error}$

• Reject $H_0$ if $p<\alpha$

• Fail to Reject $H_0$ if $p > \alpha$

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

• 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

$H_0 : \mu_1 = \mu_2$

$H_A : \mu_1 \ne \mu_2$

• Sample Averages

• 8.94 and 11.14
• Size of Each Sample

• 18 and 14
• Sample Standard Deviations

• 2.65 and 2.18
• Standard Error

• 0.85
• t-Statistic

• $\pm 2.58$
• $t^*$ Critical value for two-tailed test at $\alpha=0.05$

• degrees of freedom = $n_1 + n_2 - 2$
• $\pm 2.042$
• Reject the Null since $t > t*$

• Prices are significantly different for both areas

• df = pd.read_csv('./data/FoodPrices.csv')

data1 = list(df.AverageMealPriceArea1.dropna().values)
data2 = list(df.AverageMealPriceArea2.dropna().values)

n1 = len(data1) # 18
n2 = len(data2) # 14
df = n1 + n2 - 2 # 30

print(f'n1 = {n1} and n2 = {n2}') # 18 14
print(f"df = {df}") # 30

xbar1 = np.mean(data1) # 8.94
xbar2 = np.mean(data2) # 11.14
print(f'mean1 = {xbar1:.2f} and mean2 = {xbar2:.2f}') # 8.94 11.14

# Delta Degrees of Freedom 1 for sample
std1 = np.std(data1, ddof=1) # 2.65
std2 = np.std(data2, ddof=1) # 2.18

print(f'std1 = {std1:.2f} and std2 = {std2:.2f}') # 2.65 2.18

S = np.sqrt(std1**2/n1 + std2**2/n2) #0.85
print(f'S = std error = {S:.2f}') # 0.85

t = abs((xbar1 - xbar2)/S) # direction doesn't matter
print(f't = {t:.2f}') # 2.58

alpha = 0.05/2

print(f"df={df} Alpha={alpha} t={t}")
t_critical = 2.042 # from table
print(f"from table t-critical {t_critical}")

from scipy import stats
t_critical = stats.t.ppf(1-alpha, df)
print(f't_critical = pm {t_critical:.3f}') # t_critical = pm 2.042

if t > t_critical:
print("t is greater than t-critical") # true
print("Reject Null") # true
else:
print("t is less than t-critical")
print("fail to reject null")


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