A 95% Confidence Interval Hypothesis

ANSWER

In conclusion A 95% Confidence Interval Hypothesis-Based
The following are the hypotheses for this randomized trial:

Null Hypothesis (H0): The percentage of patients who experience a pain reduction of three or more points is the same for both the new and conventional pain relievers.
0: 1 = 2
For H 0, p 1 = p 2.
​

Alternative Hypothesis (Ha): Compared to the traditional pain reliever, a greater percentage of patients experience a pain reduction of three or more points with the new medication.
𝐻𝑎: 𝑝 1 > 𝑝 2
H a: p 1 > p 2
​

Where:

p 1 = the percentage of patients who saw a 3+ point decrease in pain as a result of using the new painkiller.
𝑝 2 p 2 represents the percentage of patients whose pain was reduced by 3+ points using the conventional painkiller.
Test Results
The test statistic (z) for two independent proportions can be found using the following formula:

𝑧 = (𝑝^1 −𝑝^2)\���^( 1 −𝑝^)
( 1 𝑛 1 + 1 𝑛 2)
z = p^ (1− p^ )( n 1

1 + n 2

1 )

(p^
​

1 − p^
​

2 )

Where:

𝑝^1 = 0.46 p^
​

The new pain reliever’s success rate is 1 = 0.46, and p = 0.22.
​

The proportion of success for the typical pain reliever is 2 = 0.22, 𝑛 1 = 50 n 1 = 50, 𝑛 2 = 50 n 2 = 50 (sample sizes), 𝑝^ = 𝑥 1 + 𝑥 2 𝑛 1 + 𝑛 2 p^ = n 1 +n 2.
​

x 1 + x 2
​

(percentage of poverty).
Calculations: Pooled Proportion (p^) = 23 + 11 50 + 50 = 34 100 = 0.34 p^ = 50 + 50 23 + 11 = 100 34 = 0.34
The standard error (SE) is:
𝑆 𝐸 = 𝑝^^(1 − 𝑝²)
( 1 𝑛 1 + 1 𝑛 2)
SE is equal to p^ (1− p^ )( n 1)
​

1 + n 2

1 )

𝑆 𝐸 = 0.34 ( 1 − 0.34 ) ( 1 50 + 1 50 ) = 0.34 ( 0.66 ) ( 2 50 ) = 0.34 ⋅ 0.66 ⋅ 0.04 = 0.008976 ≈ 0.0947 SE= 0.34(1−0.34)( 50 1​+ 50 1​)​= 0.34(0.66)( 50 2​)​= 0.34⋅0.66⋅0.04​= 0.008976​≈0.0947 Test Statistic ( 𝑧 z): 𝑧 = 𝑝 ^ 1 − 𝑝 ^ 2 𝑆 𝐸 = 0.46 − 0.22 0.0947 = 0.24 0.0947 ≈ 2.53 z= SE p ^
​

1 − p^
​

2.

is equal to 0.0947 0.46 − 0.22 = 0.0947 0.24 ≈ 2.53.
Confidence Interval of 95%
The 95% confidence interval for the proportional difference (p^2)
​

1 − p^
​

2) is:

CI = (p^2) ± 𝑧∗ ⋅ 𝑆 𝐸 CI = (p^2)
​

1 − p^
​

2 )±z ∗ ⋅SE
Using z ∗ = 1.96 for a 95% confidence level and 𝑧 ∗ = 1.96:

CI is 0.24 ± 1.96 ≅ 0.0947.
CI = 0.24±1.96⋅0.0947
CI is equal to 0.24 ± 0.18856.
CI is equal to (0.0544,0.4256).
CI = (0.0544,0.4256)
In conclusion
At the 0.05 level, statistical significance is indicated by the z-statistic of 2.53, which is more than the crucial value of 1.96. The conclusion that the new pain reliever is more effective than the conventional pain reliever is further supported by the fact that the 95% confidence interval for the difference in proportions (0.0544,0.4256) does not include zero.

Ultimately, the null hypothesis should be rejected. When compared to the conventional pain reliever, the new one is noticeably more effective in reducing pain by three or more points.

Citations
L. M. Sullivan (2018). Public health biostatistics essentials (3rd ed.). Bartlett & Jones Learning.
2021; Motulsky, H. A nonmathematical manual for statistical reasoning, Intuitive Biostatistics (4th ed.). Oxford University Press.

 

 

 

QUESTION

Module 6:  Assignment 1

 

Read the information below to write a conclusion based on a 95% Confidence Interval for the following randomized trial.

 

A randomized trial is designed to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery. The trial compares the new pain reliever to the pain reliever currently in use (called the standard of care). A total of 100 patients undergoing joint replacement surgery agreed to participate in the trial. Patients were randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery and were blind to the treatment assignment. Before receiving the assigned treatment, patients were asked to rate their pain on a scale of 0-10 with higher scores indicative of more pain. Each patient was then given the assigned treatment and after 30 minutes was again asked to rate their pain on the same scale. The primary outcome was a reduction in pain of 3 or more scale points (defined by clinicians as a clinically meaningful reduction). The following data were observed in the trial.

Treatment Group	n	Number with Reduction of 3+ Points	Proportion with Reduction of 3+ Points
New Pain Reliever	50	23	0.46
Standard Pain Reliever	50	11	0.22

 

Hint 1: Tests with Two Independent Samples, Dichotomous Outcome.

Hint 2: Remember the Confidence Interval z score table.

 

· Contribute a minimum of (1) page listing the hypotheses, test statistic used, standard z score, and your conclusion based on your calculations.

 

Submission Instructions:

· Submit your assignment by 11:59 pm on Sunday.

· It should include at least 2 academic sources, formatted and cited in APA.