Hypothesis Testing: One Sample Test (Part 2) - LO9

LO9 Hypothesis Testing: One Simple Test (Part 2)

Visualizing Hypothesis Testing

Graphical Representation: The process involves comparing a test statistic to critical values to decide whether to reject the null hypothesis (Ho) or not.
- Reject Ho regions are located in the tails of the distribution.
- Do not reject Ho region is typically in the center of the distribution.
- Lower critical value (e.g., $-t{\alpha/2}$ ) and Upper critical value (e.g., $+t{\alpha/2}$ ) define the boundaries of the rejection regions.
Hypotheses:
- Null Hypothesis (Ho): Always includes an equal sign ( $=\text{??}$ ).
- Alternative Hypothesis (H1 or Ha): Specifies the condition being tested (e.g., $\neq$ , <, >, referring to the population parameter.).

LO9.01 One-Tail Tests

Characteristics of One-Tail Tests:
- There is only one critical value because the rejection area is in only one tail of the distribution.
- The equal sign ( $=$ , $\geq$ , or $\leq$ ) always stays with the null hypothesis (Ho).
Formulating One-Tail Hypotheses:
- Testing if the population mean is less than 55:
 - $H_0: \mu \geq 55$
 - H_1: \mu < 55
- Testing if the population mean is more than 55:
 - $H_0: \mu \leq 55$
 - H_1: \mu > 55
Performing a One-Tailed T-Test using Minitab:
- Navigate to: Stat > Basic Statistics > 1-sample t...
Finding T-Critical Values for One-Tail Tests using Minitab:
- Navigate to: Graph > Probability Distribution Plot... > View Probability

LO9.2 Hypothesis Testing Proportions (Two-Sided)

Hypotheses for Two-Sided Proportion Test:
- Null Hypothesis: $H0: \pi = \pi0$
- Alternative Hypothesis: $H1: \pi \neq \pi0$
- Where $\pi$ is the population proportion and $\pi_0$ is the hypothesized population proportion.
Three Approaches to Hypothesis Testing for Proportions:
1. Critical Value Approach:
 - Test Statistic ( $z{stat}$ ): Calculated as $z{stat} = \frac{p - \pi0}{\sqrt{\frac{\pi0(1-\pi_0)}{n}}}$
 - Where:
 - $p = X/n$ is the sample proportion (number of successes $X$ out of sample size $n$).
 - $\pi_0$ is the hypothesized population proportion.
 - $n$ is the sample size.
 - Always use the z (normal approximation) for proportions, provided the conditions are met.
 - Finding Critical Values in Minitab:
 - Navigate to: Graph > Probability Distribution Plot > View Probability.
 - Select Distribution: Normal (0,1).
 - In Shaded Area, choose Both tails and input the significance level $\alpha$ .
 - The Cutoffs shown will be $\pm z_{critical}$ .
 - Decision Rule: Reject Ho if |z{stat}| > z{critical}.
2. p-Value Approach:
 - Understanding p-Value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.
 - Finding p-Value in Minitab:
 - Use the same menu as for critical values.
 - In Shaded Area, choose X value(s) Both tails and input the calculated $z_{stat}$ .
 - Minitab shades both tails beyond $|z_{stat}|$ , and the total shaded area represents the p-value.
 - Decision Rule: Reject Ho if p-value < \alpha (the significance level).
3. Confidence Interval (CI) Approach:
 - Confidence Interval Formula: The $(1-\alpha) \times 100\%$ confidence interval for the population proportion $\pi$ is given by $p \pm z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$
 - Note: This formula uses the sample proportion $p$ for the standard error in the CI calculation.
 - Decision Rule: Reject Ho if the hypothesized proportion $\pi_0$ is outside this calculated Confidence Interval.
Performing Hypothesis Testing for Proportion from Minitab:
- Navigate to: Stat > Basic Statistics > 1 Proportion...
- Condition for using this option: Ensure that $np \geq 5$ and $n(1-p) \geq 5$ . If these conditions are met, the normal approximation for proportions is valid.

L09.3 Possible Errors in Hypothesis Test Decision Making

Understanding Potential Outcomes: When performing a hypothesis test, there are four possible outcomes based on the actual truth of Ho and your decision.
- Actual Situation: Ho True
  - Decision: Do Not Reject Ho
    - Outcome: No Error
    - Probability: $1 - \alpha$
  - Decision: Reject Ho
    - Outcome: Type I Error
    - Probability: $\alpha$ (This is the Significance Level)
- Actual Situation: Ho False
  - Decision: Do Not Reject Ho
    - Outcome: Type II Error
    - Probability: $\beta$
  - Decision: Reject Ho
    - Outcome: No Error
    - Probability: $1 - \beta$ (This is the Power of the test)
Key Definitions:
- Confidence Level: The confidence level of a hypothesis test is expressed as $(1-\alpha) \times 100\%$ . It represents the probability of correctly not rejecting a true null hypothesis.
- Power of a Statistical Test: The power of a statistical test is defined as $(1 - \beta)$ . It is the probability of correctly rejecting the null hypothesis (Ho) when it is actually false.

LO9 Hypothesis Testing: One Simple Test (Part 2)

Visualizing Hypothesis Testing: Getting to the Goal!

What's the Big Idea? We're trying to make a smart guess about a big picture (the whole population) based on a small snapshot (our sample data). The "goal" is to decide whether our snapshot is so different that it proves our initial assumption about the big picture is wrong.
Imagine a Target!
- At the center of the target is where you'd expect your sample results to land if your basic assumption (the Null Hypothesis, Ho) is true. This is the "Do not reject Ho" zone.
- The edges (or tails) of the target are the "Reject Ho" regions. If your sample result lands way out here, it's so far from what's expected under Ho that you decide Ho must be false.
- Lower critical value (e.g., $-t{\alpha/2}$ ) and Upper critical value (e.g., $+t{\alpha/2}$ ) are like the boundaries defining these "danger zones" in the tails.
Meet the Hypotheses: Your Assumptions and Your Challenge
- Null Hypothesis (Ho): This is your default assumption, always includes an equal sign ( $=\text{??}$ ). Think of it as "innocent until proven guilty." This is what you try to disprove.
- Alternative Hypothesis (H1 or Ha): This is your challenge to the default. It states what you believe to be true (e.g., $\neq$ , <, >, referring to the population parameter.). This is what you're trying to prove.

LO9.01 One-Tail Tests: Looking for a Specific Direction

What's the Goal Here? You're not just asking "Is it different?" but rather "Is it specifically less than?" or "Is it specifically greater than?" You have a clear directional hunch.
Think of a One-Way Street!
- There's only one critical value because your "danger zone" (rejection area) is concentrated entirely in one tail of the distribution. You only care if the value is too high OR too low, not both.
- The equal sign ( $=$ , $\geq$ , or $\leq$ ) always stays with the Null Hypothesis (Ho). Ho gives us the starting point, H1 points in the direction we're investigating.
Formulating One-Tail Hypotheses: Setting Your Direction
- If you suspect the population mean is less than 55:
 - Ho: The mean is 55 or more ( $H_0: \mu \geq 55$ )
 - H1: The mean is definitely less than 55 (H_1: \mu < 55)
- If you suspect the population mean is more than 55:
 - Ho: The mean is 55 or less ( $H_0: \mu \leq 55$ )
 - H1: The mean is definitely more than 55 (H_1: \mu > 55)
Doing a One-Tailed T-Test in Minitab:
- Go to: Stat > Basic Statistics > 1-sample t...
Finding the t-Critical Values in Minitab for One-Tail:
- Go to: Graph > Probability Distribution Plot... > View Probability

LO9.2 Hypothesis Testing Proportions (Two-Sided): Just "Is It Different?"

What's the Goal Here? We want to know if the proportion of something in the population is just plain different from a specific value. We don't care if it's higher or lower, just that it's not what we hypothesized.
The Hypotheses for a Two-Sided Proportion Test: Two Directions of "Different"
- Null Hypothesis (Ho): The population proportion is exactly a certain value ( $H0: \pi = \pi0$ )
- Alternative Hypothesis (H1): The population proportion is not equal to that value ( $H1: \pi \neq \pi0$ )
 - Here, $\pi$ is the true population proportion, and $\pi_0$ is the hypothesized proportion you're testing against.
Three Ways to Make Your Decision: Different Tools for the Same Goal
1. Critical Value Approach: Is Your Sample "Extreme Enough"?
 - Think of Warning Lines! You calculate a Test Statistic ( $z_{stat}$ ) for your sample. This is your sample's "score." Then you compare it to critical values ( $\pm z_{critical}$ ), which are like pre-set warning lines. If your score goes beyond these lines, it's extreme enough to reject Ho.
 - How to Calculate Your Score ( $z{stat}$ ): \z{stat} = \frac{p - \pi0}{\sqrt{\frac{\pi0(1-\pi_0)}{n}}} where:
 - $p = X/n$ is your sample's proportion (number of successes $X$ out of sample size $n$).
 - $\pi_0$ is the proportion you hypothesized in Ho.
 - $n$ is your sample size.
 - Important Note: Always use the z (normal approximation) for proportions if you meet certain conditions ( $np \geq 5$ and $n(1-p) \geq 5$ ).
 - Finding the Warning Lines ( $\pm z_{critical}$ ) in Minitab:
 - Go to: Graph > Probability Distribution Plot > View Probability.
 - Select Distribution: Normal (0,1).
 - In Shaded Area, choose Both tails and input your chosen significance level, $\alpha$ (e.g., 0.05).
 - Minitab will show the Cutoffs as $\pm z_{critical}$ (your warning lines!).
 - Decision Rule: Reject Ho if the absolute value of your sample's score ( $|z{stat}|$ ) is greater than the warning line ( $z{critical}$ ).
2. p-Value Approach: How "Surprising" Is Your Sample if Ho were True?
 - The "Surprise Factor"! The p-value tells you: "If Ho (your default assumption) were actually true, how likely would it be to get a sample result as extreme, or even more extreme, than the one I just got?"
 - Finding the "Surprise Factor" in Minitab:
 - Use the same menu as for critical values (Probability Distribution Plot).
 - In Shaded Area, choose X value(s) Both tails and input your calculated $z_{stat}$ (the sample's score).
 - Minitab will shade both tails beyond $\pm |z_{stat}|$ and the total shaded area is your p-value.
 - Decision Rule: Reject Ho if your p-value (the "surprise factor") is smaller than your chosen significance level, $\alpha$ (your allowed level of risk for being surprised).
3. Confidence Interval (CI) Approach: Does Your "Plausible Range" Include Ho?
 - The "Net" Approach! You calculate a confidence interval, which is a range of values where you're pretty confident the true population proportion lies. Then you ask: "Does our hypothesized proportion (from Ho) fall inside this plausible range, or is it outside?"
 - Confidence Interval Formula: The $(1-\alpha) \times 100\%$ confidence interval for the population proportion $\pi$ is given by $p \pm z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$ (Note: this formula uses your sample's proportion $p$ for the standard error).
 - Decision Rule: Reject Ho if your hypothesized proportion $\pi_0$ is outside this calculated Confidence Interval (i.e., your "net" didn't catch it).
Performing Proportion Hypothesis Testing from Minitab:
- Go to: Stat > Basic Statistics > 1 Proportion...
- Condition: Be sure that $np \geq 5$ and $n(1-p) \geq 5$ are met. If these "safety checks" pass, the normal approximation for proportions is valid.

L09.3 Possible Errors in Hypothesis Test Decision Making: Oops! (The Risks)

The Goal of Understanding Errors: No decision based on a sample is 100% perfect. It's crucial to understand the types of mistakes you could make and their probabilities.
Think of a Courtroom Jury!
- Actual Situation: Ho is True (The defendant is truly innocent)
  - Your Decision: Do Not Reject Ho (You correctly find them innocent)
    - Outcome: No Error!
    - Probability: $1 - \alpha$ (Confidence Level - the chance of being right when Ho is true)
  - Your Decision: Reject Ho (You wrongly find them guilty)
    - Outcome: Type I Error (Convicting an innocent person!)
    - Probability: $\alpha$ (This is your Significance Level - the risk you accept for this type of error)
- Actual Situation: Ho is False (The defendant is truly guilty)
  - Your Decision: Do Not Reject Ho (You wrongly let them go free)
    - Outcome: Type II Error (Letting a guilty person walk free!)
    - Probability: $\beta$
  - Your Decision: Reject Ho (You correctly find them guilty)
    - Outcome: No Error!
    - Probability: $1 - \beta$ (This is the Power of your test - the chance of correctly catching a guilty person)
Key Definitions: Naming the Risks and Successes
- Confidence Level: Expressed as $(1-\alpha) \times 100\%$ It's the probability of correctly not rejecting a true null hypothesis. (Correctly identifying an innocent person).
- Power of a Statistical Test: Defined as $(1 - \beta)$ . It's the probability of correctly rejecting the null hypothesis (Ho) when it is actually false. (Correctly identifying a guilty person).