| Title | Author | Created | Published | Tags | | ------------------ | ---------- | ----------------- | ----------------- | -------------------------------------------------- | | Hypothesis Testing | Jon Marien | November 06, 2025 | November 06, 2025 | [[#classes\|#classes]], [[#MATH26367\|#MATH26367]] | # Hypothesis Testing ## Outline - Everyday Example - What is hypothesis testing? - General Idea of Hypothesis Testing - Generating Hypotheses - Making Decisions about the Null Hypothesis - Errors in Hypothesis Testing ### Everyday Example A cell phone manufacturer has recently released a new model of their phone and one of the key marketing features is an adapter that allows the phone to charge in 10 minutes or less once not in use or turned off. `Charge time ≤ 10 mins` Customer Review: A customer purchases the new model and expresses their disappointment that the phone is taking longer than 10 minutes to charge. Other customers have written similar reviews and this has come to the attention of the media. `Charge time > 10 mins` Manufacturer: Minutes to charge phone ≤10 minutes Customer: Minutes to charge > 10 minutes What can be done to address the customers’ concerns? `Charge time ≤ 10 mins` An independent researcher is hired to investigate the claims by the cell phone manufacturer that the length of time to charge the phone is 10 minutes or less. The researchers randomly choose a number of these cell phones from different carriers in the region to test the manufacturers claim. `Charge time ≤ 10 mins` #### Key Points - Initial claim by manufacturer: Charge time ≤ 10 mins - Counter claim by customer : Charge time > 10 mins - Question: Is the charge time different from the manufacturer’s claim? - Collect evidence: Charge time for samples of cell phones were tested - Decision: Based on evidence collected a decision is made about whether or not the manufacturer’s claim is supported or not ### What is Hypothesis Testing? - Hypothesis testing is a statistical procedure that allows us to use data from a sample to draw inferences about a population. ![[image-1006.png]] ### General Idea (of Hypothesis Testing) - Make an initial assumption about the population - THEN, HYPOTHESIS - Collect evidence to test this assumption - THEN, DATA - Based on the evidence collected decide whether or not the initial assumption is likely or unlikely - THEN, PROBABILITY ![[image-1009.png]] ## Generating Hypotheses - Two hypotheses are stated about the population of interest: - Null Hypothesis ($H_{0}$): - Statement of no difference, no effect, or no relationship - Alternative or Research Hypothesis ($H_{1}$): - Opposite the null hypothesis - Statement of a difference, an effect, or a relationship ### Example Research has shown that the mean intelligence score in the general adult population on a widely used intelligent quotient (IQ) test, the Wechsler Adult Intelligence Scale (WAIS) is $µ = 100$ with a population standard deviation of $σ =15$. A researcher is interested in examining whether or not the mean intelligence score for residents in a large nursing home is different from the general adult population. - Null hypothesis (No mean difference in IQ): $H_{0}:µ = 100$ - Alternative hypothesis (Mean difference in IQ): $H_{1}: µ ≠ 100$ ![[image-1018.png]] ### Example 2 Research has shown that the mean intelligence score in the general adult population on a widely used intelligent quotient (IQ) test, the Wechsler Adult Intelligence Scale (WAIS) is $µ = 100$ with a population standard deviation of $σ =15$. A researcher is interested in examining whether or not the mean intelligence score for residents in a large nursing home is **lower** than that in the general adult population. - Null hypothesis (Mean IQ **is not lower** for nursing home residents vs general adult population): $H_{0}:µ > 100$ - Alternative hypothesis (Mean IQ **is lower** for nursing home residents vs general adult population): $H_{1}: µ < 100$ ![[image-1023.png]] ## Generating Hypotheses Cont. A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. Will the pulse rate increase, decrease or remain unchanged after a patient takes the medication? Since the researcher knows that the mean pulse rate for the population under study is 82 beats per minute, the hypothesis for this situation are: $H_{0}: μ = 82$ $H_{1}: μ ≠ 82$ The null hypothesis specifies that the mean will remain unchanged, and the alternative hypothesis states that it will be different. ## Critical Value Approach: Steps 1. State the Hypothesis: - The null hypothesis is the assumption to be tested - The claim can be either the null hypothesis or the alternative one 2. Set the alpha ($α$) level and locate the critical region (unlikely values): - $α$ is the risk that a true null hypotheses will be rejected - The critical region is the part of the sampling distribution that is unlikely to contain the test statistics if $H_{0}$ is true. 3. Convert **sample statistic** (e.g., mean) to **test statistic** (e.g., z-score): - A Statistical Test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. - The numerical value calculated from the sample data in a statistical test is called the test statistic. - The z-test is a statistical test for the mean of a population. It is used when the population standard deviation, is known and the population is approximately normally distributed $ z=\frac{\overline{x}-\mu}{\sigma\sqrt{n}}$ 4. Make a decision about $H_{0}$: - If test statistic falls in critical region, **Reject** $H_{0}$ - Otherwise, **Do not reject** $H_{0} 5. **Summarize the results!** ### Critical Value Approach: Example 1. State the Hypothesis: - Null Hypothesis: $H_{0}:µ = 100$ - Alternative Hypothesis: $H1: µ ≠ 100$ - ![[image-1033.png]] 2. Set the alpha ($α$) level and locate the critical region (unlikely values): - $α = .05$ - Sample means beyond the critical boundaries of $z ± 1.96$ 3. Convert **sample statistic** (e.g., mean) to **test statistic** (e.g., z-score): $ z=\frac{\overline{x}-\mu}{\sigma\sqrt{n}}$ $ Z=\frac{M-\mu}{\sigma_M}=\frac{92-100}{15\sqrt{25}}=\frac{-8}3=-2.67$ 4. Make a decision about $H_{0}$: - $-2.67 < -1.96$: Sample mean falls in critical region - **Decision: Reject the null hypothesis** 5. **Summarize the results! - There is a difference in mean IQ scores between residents in a large nursing home and the general adult population. ## p-Value Approach - p-value means probability (ranges from 0 to 1) - p-value is the probability of obtaining a test statistic as extreme or more extreme than that found if there is no difference, no effect, or no relationship - Extreme means far from what we would expect based on $H_0$ - p-value provides information about the amount of statistical evidence that supports the null hypothesis - Smaller p-value = less support for $H_{0}$ - p-value compared to α level to make statistical decision: - If $p-value < α$: **Reject** $H_{0}$ - If $p-value > α$: **Do not reject** $H_{0}$ - **Critical value of $z = -2.67 < p = .0076$ (Reject $H_{0}$)** - Typically reported in the output from statistical software packages and research journals ## One-tailed vs Two-tailed Tests ![[image-1036.png]] ### Example A researcher reports that the average salary of full time professors in Canadian Universities is $\$100,735$. A sample of $30$ full professors has a mean of $\$103,746$. At $α = 0.05$, test the claim that full professors earn more than $\$100,735$ a year. From a prior study, the standard deviation of the population is known to be $\$12,499$. 1. State the hypotheses: 1. $H_{0}: μ = \$100,735$ and $H_{1}: μ > \$100,735$ (claim) 2. Set the alpha ($α$) level and locate the critical region - Since $α=0.05$ and the test is a right –tailed test, $z=1.645$. The critical region is to the right of $1.645$ 3. Compute the test statistic: $ z=\frac{\overline{x}-\mu}{\sigma}=\frac{103,746-100,735}{12,499}\approx1.319.$ 4. Make a decision about H0 - Since the test statistic, $1.319$, is less than the critical value of $1.645$, and is not in the critical region, **DO NOT reject the null hypothesis**. 5. Summarize the results - There is not enough evidence to support the claim that full professors earn more on average than $\$100,735$ a year.