Hypothesis Testing for Population Proportions and Means

Testing Hypothesis About Population

Definition of Hypothesis

A hypothesis is a conjecture about the nature of a population. In statistics, hypotheses are usually stated in terms of a parameter, such as the population mean (μ) or population proportion (p).

• Population proportion: $p = 0.3$

• Population mean: $μ = 8.7$

• Proportion favoring candidate: $p = 0.55$

Definition of Test of Hypothesis

A test of hypothesis is a statistical procedure used to make a decision about the conjectured value of a population parameter. This involves collecting sample data and determining whether the observed data contradicts the hypothesis.

Null and Alternative Hypotheses

• Null hypothesis ($H_0$): Specifies the value of the population parameter. The experiment is conducted to determine if the sample data contradicts $H_0$.

• Alternative hypothesis ($H_a$): Gives an opposing conjecture to $H_0$. The experiment tests whether the sample data supports $H_a$.

Example Applications

• Medical Test: Comparing a new cancer screening method to an existing one (e.g., does the new method detect cancer in a higher proportion of patients?).

• Market Research: Testing if the proportion of orange juice drinkers preferring a brand is less than a claimed value.

• Clinical Study: Determining if the proportion of patients with psychosomatic illnesses has changed from a historical value.

Key Concepts in Hypothesis Testing

Observed Significance Level (p-value)

The p-value is the probability of obtaining the observed value of the sampled statistic, or a value more supportive of the alternative hypothesis, assuming the null hypothesis is true.

• For $H_0: p = 0.30$ vs $H_a: p < 0.30$, p-value is the probability of getting a sample proportion greater than or equal to the observed value.

• For $H_0: p = 0.20$ vs $H_a: p > 0.20$, p-value is the probability of getting a sample proportion less than or equal to the observed value.

• For $H_0: p = 0.30$ vs $H_a: p eq 0.30$, p-value depends on whether the sample statistic is less than or greater than the claimed value.

Test Statistic

A test statistic is a quantity calculated from the sample data and used to compute the p-value. For testing a population proportion, the test statistic is:

$z = rac{ar{p} - p}{ ext{sqrt} rac{p(1-p)}{n}}$

Where:

• $ar{p}$ = sample proportion

• $p$ = hypothesized population proportion

• $n$ = sample size

Decision Rule

The decision rule specifies when to reject the null hypothesis. Typically:

• Accept the alternative hypothesis if the p-value is less than the significance level $α$.

• Common significance level: $α = 0.05$

• Reject $H_0$ if p-value < $α$

The Elements of a Test of Hypothesis

Element	Description
1. Null hypothesis ($H_0$)	Statement about the population parameter
2. Alternative hypothesis ($H_a$)	Opposing statement to $H_0$
3. Decision rule	Criteria for accepting/rejecting $H_0$
4. Test statistic	Calculated value from sample data
5. p-value	Probability of observed result under $H_0$
6. Interpretation	Conclusion based on the test

Identifying a z-Test of Hypothesis About p

• The population of interest contains qualitative data. The parameter of interest is the population proportion $p$.

• The experimental objective is to determine if sampled data supports the claim that $p$ is greater than, less than, or not equal to some hypothesized number.

• A large, random sample of data is collected and the sample proportion $ar{p}$ is calculated.

Elements of a z-Test of Hypothesis About μ

• Define the parameter μ in the context of the test.

• State the hypotheses:

  • $H_0: p = p_0$

  • $H_a: p > p_0$, $p < p_0$, or $p eq p_0$

• Decision Rule: Reject $H_0$ if p-value < $α$

• Test Statistic: $z = rac{ar{p} - p_0}{ ext{sqrt} rac{p_0(1-p_0)}{n}}$

• Calculate p-value, make decision, and interpret results.

Types of Errors in Hypothesis Testing

• Type I error: Concluding that the alternative hypothesis $H_a$ is true when in fact the null hypothesis $H_0$ is true. Probability of Type I error is $α$ (significance level).

• Type II error: Concluding that the null hypothesis $H_0$ is true when in fact the alternative hypothesis $H_a$ is true. Probability of Type II error is $β$.

Example: Restaurant Survey

• An entrepreneur wants to know if more than 15% of local residents are interested in a new restaurant. Survey of 500 people: 90 express interest.

• State hypotheses:

  • $H_0: p = 0.15$

  • $H_a: p > 0.15$

• Type I error: Opening the restaurant when it will not be successful.

• Type II error: Not opening the restaurant when it would have been successful.

• Choice of $α$ affects risk of errors; lower $α$ reduces risk of Type I error.

Example: Online Purchases by Graduate Students

• Claim: 60% of graduate students have made online purchases. Consumer group suspects the proportion is less than 60%.

• Sample: 80 graduate students; 44 have made online purchases.

• Parameter of interest: $p$ (proportion of graduate students who have made online purchases)

• Hypotheses:

  • $H_0: p = 0.60$

  • $H_a: p < 0.60$

• Decision rule: Reject $H_0$ if p-value < $α$

• Test statistic: $z = rac{ar{p} - 0.60}{ ext{sqrt} rac{0.60(1-0.60)}{80}}$

• Calculate p-value, make decision, and interpret results.

• Possible error: Type I error (concluding proportion is less than 60% when it is not).

Confidence Interval for a Proportion

• To estimate the proportion with 95% confidence:

Formula:

$ar{p} ext{ ± } z^* ext{sqrt} rac{ar{p}(1-ar{p})}{n}$

• $z^*$ is the critical value for the desired confidence level (for 95%, $z^* ext{ ≈ } 1.96$)

• Plug in sample values to calculate lower and upper limits.

• Interpretation: The interval gives a range of plausible values for the population proportion.

Additional info: Some context and formulas have been expanded for clarity and completeness.