This post covers Hypothesis Testing.

Hypothesis Testing

• Threshold is difficult to decide - likely or unlikely

$\alpha$ Levels of Likelihood (unlikelihood) - One Tailed

• If the probability of getting a sample mean is less than
  • $ \alpha = 0.05 (5\%)$
  • $ \alpha = 0.01 (1\%)$
  • $ \alpha = 0.001 (0.1\%)$
• then it is considered unlikely.
• If the probability of getting a particular sample mean is less than $\alpha$ (0.05, 0.01, 0.001), it is unlikely to occur
• If a sample mean has a z-score greater than $z^*$ (1.64, 2.33, 3.09), it is unlikely to occur

Z-Critical Value

• If the probability of obtaining a particular sample mean is less than the alpha level. Then it will fall in the tail which is called the Critical Region and Z-value is called the z-critical value

• If the z-score of a sample mean is greater than z-critical value, we have evidence that these sample statistics are different from regular or untreated population.

• If probability of critical region = alpha level = 0.05

  • z-critical value = 1.64

  • alpha = 5/100
    z = stats.norm.ppf(1 - alpha)
    label = f'alpha={alpha} ({alpha*100}%), z={z:.2f}'
    

• If probability of critical region = alpha level = 0.01

  • z-critical value = 2.33

  • alpha = 1/100
    z = stats.norm.ppf(1 - alpha)
    label = f'alpha={alpha} ({alpha*100}%), z={z:.2f}'
    

• If probability of critical region = alpha level = 0.001

  • z-critical value = 3.09

  • alpha = 0.1/100
    z = stats.norm.ppf(1 - alpha)
    label = f'alpha={alpha} ({alpha*100}%), z={z:.2f}'
    

• Example

  • Sample Mean = $\bar{X}$
  • $z = \frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}$
  • Let $z=1.82$
    • $\bar{X}$ is significant at $p<0.05$
    • since zcr > 1.64 (0.05) and < 2.33 (0.01)
    • Red region

• Example

  • z-score	Significant at: ( p< )
    3.14	0.001
    2.07	0.05
    2.57	0.01
    14.31	0.001

Two-Tailed Critical Values

• Split the Alpha Level in half

Hypothesis

One Tailed Test	One Tailed Test	Two Tailed Test

• Two Outcomes

  • Sample Mean is outside the Critical Region
  • Sample Mean is inside the Critical Region

• $H_0$, Null Hypothesis

  • No Significant difference between the current population parameters and what will be the new population parameters after intervention
  • Sample Mean lies outside the critical region
  • $\mu \sim \mu_I$

• $H_a$ or $H_1$, Alternate Hypothesis

  • $\mu < \mu_I$
  • $\mu > \mu_I$
  • $\mu \ne \mu_I$

• Example

  • $H_0$: Most dogs have four legs (most = more than 50%)
  • $H_A$: Most dogs have less than four dogs
  • Sample 10 dogs and find all have four legs
    • Did we prove that Null Hypothesis is True?
      • No
        • We have evidence to suggest that most dogs have four legs - since we have sample - but we didn’t prove - we also didn’t prove alternative hypothesis
        • We simply fail to reject the Null Hypothesis
  • Sample 10 dogs and find that 6 dogs have 3 legs
    • Is this evidence to reject the null hypotheis that most dogs have 4 legs
    • Yes
      • Based on sample, reject Null in favor of Alternative

• Example

  • EngagementLearning.csv

  • $\mu = 7.47, \sigma = 2.413$

  • Hypothesis Test

    • $H_0$ - no significant difference
      • not make learners more engaged
      • results in same level of engagement
    • $H_1$ - significant difference
      • Make Learners more Engaged, $\mu < \mu_I$
      • Make Learners less Engaged, $\mu > \mu_I$
      • Change how much learners are engaged, $\mu \ne \mu_I$

  • Which Hypothesis Test to choose

    • $\mu < \mu_I$ - One Tailed Test (cr - right)
    • $\mu > \mu_I$ - One Tailed Test (cr - left)
    • $\mu \ne \mu_I$ - Two Tailed Test

  • Two Tailed Test on Learning at 5 % Level

    • z-critical values, $\pm 1.96$

    • lb = round(stats.norm.ppf(0.025), 3) # -1.96
      ub = round(stats.norm.ppf(0.025+.95), 3) # 1.96
      

  • mu = 7.47
    sigma = 2.413
        
    n = 30
    xbar = 8.3
        
    std_error = sigma / np.sqrt(n)
        
    # z-score of the sample mean on the sampling distribution
    z = (xbar - mu)/std_error # 1.884
    

  • At $\alpha = 0.05$, do we reject or fail to reject the null

    • Fail to reject since z-score is less than critical value so it is outside the critical region - fail to reject the null hypothesis
    • Not enough evidence that the new population parameters will not be significantly different than the current

  • mu = 7.47
    sigma = 2.413
        
    # large sample size
    n = 50
    xbar = 8.3
        
    std_error = sigma / np.sqrt(n)
    # z-score of the sample mean on the sampling distribution
    z = (xbar - mu)/std_error # 2.43
    z
    

  • At $\alpha = 0.05$, do we reject or fail to reject the null

    • Reject the Null Hypothesis, $p < 0.05$
    • Enough evidence that the new population parameters will be significantly different than the current

  • What is the probability of randomly selecting a sample of size 50 with mean of at least 8.3 from the population

    • mu = 7.47
      sigma = 2.413
            
      n = 50
      xbar = 8.3
      std_error = sigma / np.sqrt(n)
      z = (xbar - mu)/std_error
      print(z) # 2.43
            
      p = 1 - round(stats.norm.cdf(z), 3) 
      print(p) # 0.008
      

Decision Errors

• Example
  • $H_0$: The beverage is fine to drink now
  • $H_A$: The beverage is too hot to drink
    • A - you decide the beverage is fine to drink now, but it’s too hot and you burn your tongue
      • Retain $H_0$ and False $H_0$
    • B - You decide the beverage is fine to drink now, and it is
      • Retain $H_0$ and True $H_0$
    • C - Yoy think the beverage is too hot so you wait to drink it, but it’s actually fine now and by the time you drink it, its too cold
      • Reject $H_0$ and True $H_0$
    • D - You think the beverage is too hot and indeed it is, so you wait to drink it and then it’s perfect
      • Reject $H_0$ and False $H_0$
• Example
  • $H_0$: Its not going to rain
  • $H_A$: It will rain
    • A - It doesn’t rain
      • $H_0$ True
    • B - You didn’t bring your umbrella
      • Retain $H_0$
    • C - You bring your umbrella
      • Reject $H_0$
    • D - It rains
      • $H_0$ False

Learning Example

$H_0: \mu_I = \mu$

$H_A: \mu_I \ne \mu$

$\mu = 7.47,~ \sigma = 2.41$

$n=30,~ \bar{X}=8.3, \mu_{new} =7.8$

Two-tailed Test $\alpha=0.05$

Ans: Retain $H_0$ and $H_0$ is correct

mu = 7.47
std = 2.41

n = 30
xbar = 8.3
mu_new = 7.8

std_error = std/np.sqrt(n)

z = (xbar - mu)/std_error
z = round(z, 3)
print(z) # 1.886

print('z-critical at 0.05 is 1.96')
print('Retain Null')

z = (mu_new - mu)/std_error
z = round(z, 3)
print(z) # 0.75
print('H0 is True since mu_new is not significantly different from mu since mu_new is outside critical region')

print('Retain H0 when H0 is True')
print('Correct')

• Sample Size = 50

mu = 7.47
std = 2.41

n = 50
xbar = 8.3
mu_new = 7.8

std_error = std/np.sqrt(n)

z = (xbar - mu)/std_error
z = round(z, 3)
print(z) # 2.435

print('z-critical at 0.05 is 1.96')
print('Reject Null')

z = (mu_new - mu)/std_error
z = round(z, 3)
print(z) # 0.968
print('H0 is True since mu_new is not significantly different from mu since mu_new is outside critical region')

print('Reject H0 when H0 is True')
print('Type I Error')