Comprehensive Guide to Hypothesis Testing: Critical Value and P-Value Methods

Foundations of Hypothesis Testing

• Fundamental Requirements for Running a Hypothesis Test:

  • Sample data collected randomly from the population under study.
  • A worded claim, typically presented in a problem statement or narrative paragraph, which must be converted into symbolic form.
  • A defined null hypothesis and alternative hypothesis.
  • Identification of which hypothesis represents the claim.
  • A statistical test (test formula) appropriate for the parameter and conditions being analyzed.

• The Hypotheses and the Claim:

  • Null Hypothesis ($H_0$): Always contains the equal sign ($=$). It represents the statement of no effect, status quo, or the hypothesized parameter value.
  • Alternative Hypothesis ($H_1$ or $H_a$): Contains a directional or non-directional comparison symbol: strictly less than ($<$), strictly greater than ($>$), or not equal to ($\neq$).
  • Locating the Claim:
  • The claim can reside in either the null hypothesis or the alternative hypothesis depending on the wording of the problem.
  • If the statement specifies that a parameter "is equal to" or simply "is" a certain value, the claim is the null hypothesis ($H_0$).
  • If the statement specifies that a parameter is "greater than", "less than", or "different from" (not equal to) a certain value, the claim is the alternative hypothesis ($H_1$).

• The Test Statistic and Decision Rationale:

  • A statistical test generates a single calculated numerical value termed the test statistic.
  • The formula directly compares the hypothesized population parameter to the observed sample statistic.
  • If the difference between the observed sample statistic and the hypothesized parameter is small, it is considered non-significant and attributable to chance, leading to the decision to fail to reject the null hypothesis.
  • If the difference between what is observed in the sample and what is hypothesized is large enough to be statistically significant, the decision is to reject the null hypothesis.

• The Analogy of the Null and Alternative Hypotheses:

  • The null hypothesis can be conceptualized as an existing spouse (the husband), representing the established status quo.
  • The alternative hypothesis represents an outside party (the boyfriend).
  • To accept or support the alternative hypothesis, one must first "divorce" (reject) the null hypothesis.
  • As long as the null hypothesis is maintained (failed to be rejected), one cannot move to support the alternative hypothesis.

Critical Values, Critical Regions, and Decision Rules

• Definitions of Regions and Thresholds:

  • Critical Value: A specific cutoff score (such as a standard normal $z$-score) that separates the rejection region from the non-rejection region. Critical values reside in the tails of the distribution and never in the center.
  • Critical Region (Rejection Region): The extreme tail area under the standard normal distribution curve. When a calculated test statistic falls within this region, the difference is statistically significant, requiring rejection of the null hypothesis ($H_0$).
  • Noncritical Region (Non-rejection Region): The central area of the distribution between the critical boundaries. When a calculated test statistic falls within this region, the difference is small and attributable to chance, requiring failure to reject the null hypothesis ($H_0$).

• Significance Level ($\alpha$):

  • The Greek letter alpha ($\alpha$) represents the significance level.
  • Mathematically, $\alpha$ represents both the total area allocated to the critical rejection region(s) and the maximum allowable probability of committing a Type I error.

• Types of Tests Based on Directionality:

  • Right-Tailed Test:
  • Occurs when the alternative hypothesis contains the greater-than symbol ($>$).
  • The entire area of $\alpha$ is located in the extreme right tail of the standard normal distribution curve.
  • The critical value $z$ is positive ($+z$).
  • Critical values to the right identify outcomes that are significantly high.
  • Left-Tailed Test:
  • Occurs when the alternative hypothesis contains the less-than symbol ($<$).
  • The entire area of $\alpha$ is located in the extreme left tail of the standard normal distribution curve.
  • The critical value $z$ is negative ($-z$).
  • Critical values to the left identify outcomes that are significantly low.
  • Two-Tailed Test:
  • Occurs when the alternative hypothesis contains the not-equal symbol ($\neq$).

• The area of $\alpha$ is divided equally between the two tails ($\frac{\alpha}{2}$ in the left tail and $\frac{\alpha}{2}$ in the right tail).

• Yields two distinct critical values symmetric about the mean: $-z$ and $+z$.

Procedures for Finding Critical Values Using StatCrunch

• Accessing the Normal Distribution Calculator:

  • Navigation path: Select Stat $\rightarrow$ Calculators $\rightarrow$ Normal.
  • The standard normal distribution defaults to a mean of $0$ and a standard deviation of $1$.

• Left-Tailed Test Configuration:

  • Set the inequality sign in the calculator to less than or equal to ($\le$).
  • Enter the significance level value ($\alpha$) into the area/probability box on the right side of the equals sign.
  • Click Compute.
  • The calculator displays a negative critical $z$-value.

• Right-Tailed Test Configuration:

  • Set the inequality sign in the calculator to greater than or equal to ($\ge$).
  • Enter the significance level value ($\alpha$) into the area/probability box.
  • Click Compute.
  • The calculator displays a positive critical $z$-value.

• Two-Tailed Test Configurations (Two Alternative Methods):

  • Method 1 (Between Mode):
  • Select the Between option in the Normal Calculator.
  • Calculate the central area: $1 - \alpha$.
  • For example, if $\alpha = 0.01$, compute $1 - 0.01 = 0.99$.
  • Enter the central area ($0.99$) into the area box and click Compute.
  • The calculator immediately generates both the negative and positive critical values ($-z$ and $+z$).
  • Method 2 (Tail Division Mode):
  • Divide the significance level by two: $\frac{\alpha}{2}$.
  • For an alpha of $0.01$, the split tail area is $0.005$ (noting that $0.01 / 2 = 0.005$).
  • Enter the divided tail area into the probability box for either the left tail ($\le$) or right tail ($\ge$).
  • The resulting magnitude provides the paired critical values: $\pm z$.

Decision Making and Types of Errors

• Connection Between Decision and Conclusion:

  • When the claim is the alternative hypothesis ($H_1$):
  • If the decision is to reject $H_0$, there is sufficient evidence to support the claim.
  • If the decision is to fail to reject $H_0$, there is not enough evidence to support the claim.
  • When the claim is the null hypothesis ($H_0$):
  • If the decision is to reject $H_0$, there is sufficient evidence to reject the claim.
  • If the decision is to fail to reject $H_0$, there is not enough evidence to reject the claim.

• Classification of Errors in Hypothesis Testing:

  • Type I Error:
  • Occurs when the null hypothesis ($H_0$) is rejected when it is actually true.
  • If an analysis results in rejecting $H_0$, the only potential error that could be made is a Type I error.
  • Type II Error:
  • Occurs when failing to reject the null hypothesis ($H_0$) when it is actually false.
  • If an analysis results in failing to reject $H_0$, the only potential error that could be made is a Type II error.

Traditional Method: Hypothesis Testing for the Mean ($\sigma$ Known)

• Conditions and Assumptions:

  • The population standard deviation ($\sigma$) must be known.
  • The sample size must be greater than thirty ($n > 30$), or the underlying population variable must be normally distributed.

• The Test Statistic Formula:

  • The $z$ test statistic is calculated as:

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

• Where:

  • $\bar{x}$ = Sample mean

  • $\mu$ = Hypothesized population mean stated in $H_0$

  • $\sigma$ = Population standard deviation

  • $n$ = Sample size

  • The Five-Step Traditional Hypothesis Testing Framework:

• Step 1: State the null and alternative hypotheses in symbolic form and identify the claim.

• Step 2: Find the critical value(s) from the standard normal distribution using $\alpha$ and the directionality of the test.

• Step 3: Compute the test statistic using the statistical formula.

• Step 4: Make the decision to reject or fail to reject the null hypothesis by comparing the test statistic to the critical value(s).

• Step 5: Formulate the non-technical conclusion addressing the original claim.

Example 1: Right-Tailed Test (Automobile Dealership Lot Duration)

• Problem Statement:

  • A researcher wishes to see if the mean number of days that a basic, low-price small automobile sits on a dealer lot is greater than $29$ days.
  • A sample of $30$ automobiles has a mean of $30.1$ days.
  • The population standard deviation is $3.8$ days.
  • Test the claim at $\alpha = 0.05$.

• Step-by-Step Analysis:

  • Step 1: State Hypotheses and Identify Claim:
  • $H_0: \mu = 29$
  • $H_1: \mu > 29$ (Claim)
  • This represents a right-tailed test.
  • Step 2: Determine Critical Value:
  • Right-tailed test with $\alpha = 0.05$.
  • Placing all area ($0.05$) in the right tail of the standard normal distribution yields a critical value of $z = +1.65$.
  • Critical region: Any test statistic value $z > 1.65$.
  • Step 3: Compute Test Statistic:
  • Hypothesized mean $\mu = 29$
  • Sample mean $\bar{x} = 30.1$
  • Population standard deviation $\sigma = 3.8$
  • Sample size $n = 30$
  • Computation:

$z = \frac{30.1 - 29}{\frac{3.8}{\sqrt{30}}} = \frac{1.1}{0.6938} = 1.59$

• Step 4: Make Decision:

  • The test statistic $z = 1.59$ is less than the critical value $z = 1.65$.
  • The value $1.59$ falls in the noncritical (non-rejection) region.
  • Decision: Fail to reject the null hypothesis ($H_0$).

• Step 5: State Conclusion:

  • There is not enough evidence to support the claim that the mean number of days a low-price small automobile sits on the lot is greater than $29$ days.

  • Potential error: Because $H_0$ was not rejected, the possible error committed is a Type II error.

  • StatCrunch Implementation:

• Path: Stat $\rightarrow$ Z Stats $\rightarrow$ One Sample $\rightarrow$ With Summary.

• Input values:

  • Sample mean: $30.1$
  • Standard deviation: $3.8$
  • Sample size: $30$
  • Perform: Hypothesis test for mu
  • H_0: mu =: $29$
  • H_a: mu >: $29$

• Output: Reports $z\text{-stat} = 1.59$.

Example 2: Left-Tailed Test Setup

• General Principles of a Left-Tailed Test:

  • The claim indicates that a mean is less than a hypothesized value ($H_1: \mu < \dots$).
  • Null hypothesis: $H_0: \mu = \dots$
  • Alternative hypothesis: $H_1: \mu < \dots$ (Claim).
  • Significance level used in example: $\alpha = 0.10$.

• Critical Region and Test Statistic Characteristics:

  • All of the $\alpha = 0.10$ area is dumped into the far left tail of the normal distribution.
  • The resulting critical value is negative.
  • For a left-tailed test, both the critical value and the relevant rejection test statistics reside on the negative side of zero.

• Decision and Conclusion Dynamics:

  • If the calculated test statistic is smaller (more negative) than the critical value, it falls directly into the critical region.
  • Decision: Reject the null hypothesis ($H_0$).
  • Conclusion: Because $H_0$ is rejected and the claim is in the alternative hypothesis ($H_1$), there is sufficient evidence to support the claim.
  • Potential error: Because $H_0$ was rejected, the possible error committed is a Type I error (rejecting $H_0$ when it is true).

Example 3: Two-Tailed Test (Stroke Rehabilitation Cost)

• Problem Statement:

  • The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is \twentyfourthousand six hundred seventy-two dollars ($\$24,672$).
  • To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of $35$ stroke victims at the hospital and finds that the average cost of their rehabilitation is \twentysix thousand three hundred forty-three dollars ($\$26,343$).
  • The standard deviation of the population is \threethousand two hundred fifty-one dollars ($\$3,251$).
  • At $\alpha = 0.01$, can it be concluded that the average cost of stroke rehabilitation at this particular hospital is different from $\$24,672$?

• Step-by-Step Analysis:

  • Step 1: State Hypotheses and Identify Claim:
  • The phrase "different from" denotes a non-directional difference ($\neq$).

• $H_0: \mu = 24672$

• $H_1: \mu \neq 24672$ (Claim)

  • This represents a two-tailed test.

• Step 2: Determine Critical Values:

  • Significance level $\alpha = 0.01$.
  • Divide alpha equally into two tails: $\frac{\alpha}{2} = \frac{0.01}{2} = 0.005$.
  • Using the standard normal distribution, the critical values with an area of $0.005$ in each tail are $z = -2.58$ and $z = +2.58$.
  • Critical regions: $z < -2.58$ or $z > +2.58$.

• Step 3: Compute Test Statistic:

  • Hypothesized mean $\mu = 24672$
  • Sample mean $\bar{x} = 26343$
  • Population standard deviation $\sigma = 3251$
  • Sample size $n = 35$
  • Computation:

$z = \frac{26343 - 24672}{\frac{3251}{\sqrt{35}}} = \frac{1671}{549.52} = 3.04$

• Step 4: Make Decision:

  • The test statistic $z = 3.04$ is greater than the positive critical value of $+2.58$.
  • The value falls in the upper critical (rejection) region.
  • Decision: Reject the null hypothesis ($H_0$).

• Step 5: State Conclusion:

  • There is sufficient evidence to support the claim that the average cost of rehabilitation for stroke victims at the particular hospital is different from $\$24,672$.

  • StatCrunch Implementation:

• Path: Stat $\rightarrow$ Z Stats $\rightarrow$ One Sample $\rightarrow$ With Summary.

• Inputs:

  • Sample mean: $26343$
  • Standard deviation: $3251$
  • Sample size: $35$
  • Hypothesis test: H_0: mu = 24672, H_a: mu != 24672

• Output: Displays test statistic $z\text{-stat} = 3.04$.

The P-Value Method for Hypothesis Testing

• Conceptual Definition of P-Value:

  • The P-value (probability value) is the probability of obtaining a sample test statistic as extreme as, or more extreme than, the observed test statistic, assuming that the null hypothesis is true.
  • Graphically, it is the tail area extending from the test statistic outward to the extreme end of the distribution curve.

• Comparison Rule for P-Value versus Alpha ($\alpha$):

  • If $P\text{-value} \le \alpha$, reject the null hypothesis ($H_0$).
  • If $P\text{-value} > \alpha$, fail to reject the null hypothesis ($H_0$).

• Traditional Method vs. P-Value Method Comparison:

  • The traditional method compares score metrics: the calculated test statistic ($z$-score) versus the critical value ($z$-score).
  • The P-value method compares area/probability metrics: the calculated tail probability ($P$-value) versus the significance level ($\alpha$).

Example 4: P-Value Method (College Tuition and Fees)

• Problem Statement:

  • A researcher wishes to test the claim that the average cost of tuition and fees at a four-year college is greater than \fivethousand seven hundred dollars ($\$5,700$).
  • A random sample of $36$ students has a mean of \fivethousand nine hundred fifty dollars ($\$5,950$).
  • The population standard deviation is \sixhundred fifty-nine dollars ($\$659$).
  • Is there evidence to support the claim at $\alpha = 0.05$?

• Step-by-Step Analysis:

  • Step 1: State Hypotheses and Identify Claim:
  • $H_0: \mu = 5700$
  • $H_1: \mu > 5700$ (Claim)
  • Direction: Right-tailed test.
  • Step 2: Compute Test Statistic:
  • Hypothesized mean $\mu = 5700$
  • Sample mean $\bar{x} = 5950$
  • Population standard deviation $\sigma = 659$
  • Sample size $n = 36$
  • Computation:

$z = \frac{5950 - 5700}{\frac{659}{\sqrt{36}}} = \frac{250}{109.83} = 2.28$

• Step 3: Determine the P-Value:

  • Because it is a right-tailed test, the P-value corresponds to the probability of obtaining a $z$-score greater than or equal to $2.28$: $P(z \ge 2.28)$.
  • In the standard normal calculator, evaluating the area to the right of $z = 2.28$ yields an area of $0.0113$.
  • $P\text{-value} = 0.0113$.

• Step 4: Make Decision by Comparing P-Value to $\alpha$:

  • Given $\alpha = 0.05$.
  • $P\text{-value} = 0.0113$.
  • Because $0.0113 \le 0.05$ ($P\text{-value} \le \alpha$), the decision is to reject the null hypothesis ($H_0$).

• Step 5: State Conclusion:

  • Because $H_0$ is rejected and the claim is the alternative hypothesis ($H_1$), there is sufficient evidence to support the claim that the average cost of tuition and fees at a four-year college is greater than $\$5,700$.

  • Comprehensive StatCrunch Execution for Test Statistic and P-Value:

• Navigation: Stat $\rightarrow$ Z Stats $\rightarrow$ One Sample $\rightarrow$ With Summary.

• Parameter entries:

  • Sample mean: $5950$
  • Standard deviation: $659$
  • Sample size: $36$
  • Hypothesis test: H_0: mu = 5700, H_a: mu > 5700

• Output generated:

  • Displays both the test statistic ($z\text{-stat} = 2.28$) and the exact $P\text{-value} = 0.0113$ simultaneously in a single results window.

Questions & Classroom Dialogue

• Navigating to Critical Values in StatCrunch:

  • Student Query / Classroom Prompt: Where in StatCrunch do we start and navigate to find critical values?
  • Response: Navigate through Stat to Calculators, and select Normal. Because critical values relate to the standard normal distribution, keep the mean at $0$ and standard deviation at $1$.

• Adjusting Tail Directions in Calculator:

  • Classroom Prompt: For a left-tailed test, what setting needs to be adjusted in the Normal Calculator?
  • Response: Ensure the inequality faces left ($\le$) so that the tail area is to the left of the mean, yielding a negative critical value.

• Identifying Claims in Sentence Structure:

  • Student Query: How do we know the difference between when the claim is the alternative hypothesis versus when it is the null hypothesis?
  • Response: Examine the specific wording in the problem statement. The claim is the alternative hypothesis if the statement specifies "greater than" ($>$), "less than" ($<$), or "different from / not equal to" ($\neq$). If the statement specifies that the parameter "is equal to" or "is" a certain quantity with no direction indicated, then the claim is the null hypothesis ($H_0$).

• Interpreting Test Failure vs. Claim Support:

  • Student Query: So if we fail to reject, we don't have enough evidence to support the claim? And vice versa, if we do reject, we do have enough evidence to support the claim?
  • Response: Yes, exactly, whenever the claim is the alternative hypothesis ($H_1$). If you fail to reject the null, you lack sufficient evidence to support your claim. If you reject the null, you have obtained sufficient evidence to support your claim.

• Classroom Roster Exchange:

  • Interaction Details: During the session, student reading assignments and attendance were verified, involving Logan, Trevor, Ryan, Hammond, Bradley Anderson, and Stella, regarding handwriting identification on the attendance record.