3. A fitness center is running a discounted membership fee. Did the discount increase the membership sales? Write your hypotheses mathematically.
4. A medical researcher gave 100 patients a new drug to see if it reduces their blood pressure? Did the new drug reduce the patients' blood pressure? Write your hypothesis mathematically.
5. Some students took a conflict resolution class? Did this class help to reduce the number of conflicts that the students were involved in? Write your hypothesis mathematically.
To answer these questions mathematically, we need to formulate our hypotheses. In hypothesis testing, we typically have two hypotheses: a null hypothesis and an alternative hypothesis.
3. Fitness center membership sales and discount:
Let's say the membership fee without the discount is represented by P1, and the membership fee with the discount is represented by P2. The null hypothesis (H0) would be that the discount did not increase the membership sales, so the mathematical hypothesis is:
H0: P1 = P2
The alternative hypothesis (Ha) would be that the discount did increase the membership sales, so the mathematical hypothesis is:
Ha: P1 < P2
4. New drug and blood pressure reduction:
Let's say the average blood pressure of the patients before taking the new drug is represented by μ1, and the average blood pressure of the patients after taking the new drug is represented by μ2. The null hypothesis (H0) would be that the new drug did not reduce the patients' blood pressure, so the mathematical hypothesis is:
H0: μ1 ≥ μ2
The alternative hypothesis (Ha) would be that the new drug did reduce the patients' blood pressure, so the mathematical hypothesis is:
Ha: μ1 < μ2
5. Conflict resolution class and reduction in conflicts:
Let's say the average number of conflicts that students were involved in before taking the class is represented by μ1, and the average number of conflicts the students were involved in after taking the class is represented by μ2. The null hypothesis (H0) would be that the conflict resolution class did not help reduce the number of conflicts, so the mathematical hypothesis is:
H0: μ1 ≤ μ2
The alternative hypothesis (Ha) would be that the conflict resolution class did help reduce the number of conflicts, so the mathematical hypothesis is:
Ha: μ1 > μ2
Note: These hypotheses are based on the assumption that the conditions for each study are properly defined and meet the requirements for hypothesis testing.
3. Hypotheses mathematically:
Null hypothesis (H0): The discount did not increase the membership sales.
Alternative hypothesis (H1): The discount increased the membership sales.
Mathematically, this can be written as:
H0: Δ = 0 (The mean difference in membership sales before and after the discount is zero)
H1: Δ > 0 (The mean difference in membership sales before and after the discount is greater than zero)
4. Hypothesis mathematically:
Null hypothesis (H0): The new drug does not reduce the patients' blood pressure.
Alternative hypothesis (H1): The new drug reduces the patients' blood pressure.
Mathematically, this can be written as:
H0: μ = μ0 (The mean blood pressure of patients taking the new drug is equal to the mean blood pressure without the drug)
H1: μ < μ0 (The mean blood pressure of patients taking the new drug is less than the mean blood pressure without the drug)
5. Hypothesis mathematically:
Null hypothesis (H0): The conflict resolution class does not help to reduce the number of conflicts.
Alternative hypothesis (H1): The conflict resolution class helps to reduce the number of conflicts.
Mathematically, this can be written as:
H0: p = p0 (The proportion of conflicts involving students who took the conflict resolution class is equal to the proportion of conflicts without the class)
H1: p < p0 (The proportion of conflicts involving students who took the conflict resolution class is less than the proportion of conflicts without the class)