# Hyothesis

** Published:**

This post covers Hypothesis Testing.

# Hypothesis Testing

Dependent Variable (Measured in scale from 1 to 10) | Sample Mean $\bar{X}$ (n=20) | Probability | Likely or Unlikely |
---|---|---|---|

Student Engagement | $\bar{X}_E = $ Something | $ p \sim 0.05 $ | |

Student Learning | $\bar{X}_L =$ Something | $ p \sim 0.10 $ |

- 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

- No

- Did we prove that Null Hypothesis is True?
- 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

Z | $\alpha$-Level | Test |
---|---|---|

$\pm 1.64$ | 5% or 0.05 | One tailed |

$\pm 2.33$ | 1% or 0.01 | One tailed |

$\pm 3.09$ | 0.1% or 0.001 | One tailed |

$\pm 1.96$ | 5% or 0.05 | Two tailed |

$\pm 2.58$ | 1% or 0.01 | Two tailed |

$\pm 3.29$ | 0.1% or 0.001 | Two tailed |

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, $\mu \ne \mu_I$

- $H_0$ - no significant difference
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

| | | | ———————————————————— | ———————————————————— |

Reject $H_0$ | Retain $H_0$ | |
---|---|---|

$H_0$ True | Statistical Decision Errors Type I Error | Correct |

$H_0$ False | Correct | Statistical Decision Errors Type II Error |

- 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 $H_0$

- A - you decide the beverage is fine to drink now, but it’s too hot and you burn your tongue

- 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

- A - It doesn’t rain

# 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')
```