In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse rate of patients during surgery. The available subjects were divided at random into two groups of 30 patients each. One group received a beta-blocker; the other group received a placebo. The pulse rate of each patient at a critical point during the operation was recorded. The treatment group had a mean pulse rate of 65.2 and standard dviation 7.8. For the control group, the mean pulse rate was 70.3 and the standard deviation was 8.3. What is the probability that the difference in the samples (treatment group – placebo group) is greater than 0?
To find the probability that the difference in the samples (treatment group - placebo group) is greater than 0, we will perform a hypothesis test using the t-distribution.
Step 1: State the null and alternative hypothesis:
Null hypothesis (H0): The mean difference between the treatment group and placebo group is zero.
Alternative hypothesis (H1): The mean difference between the treatment group and placebo group is greater than zero.
Step 2: Specify the significance level (alpha):
Let's assume a significance level (alpha) of 0.05. This means we want to be 95% confident in our results.
Step 3: Calculate the pooled standard deviation (Sp):
Sp = sqrt(((n1-1)*(s1^2) + (n2-1)*(s2^2))/(n1+n2-2))
Where:
n1 = size of treatment group
s1 = standard deviation of treatment group
n2 = size of placebo group
s2 = standard deviation of placebo group
Using the given values:
n1 = 30
s1 = 7.8
n2 = 30
s2 = 8.3
Sp = sqrt(((30-1)*(7.8^2) + (30-1)*(8.3^2))/(30+30-2))
Sp = sqrt((29*60.84 + 29*68.89)/58)
Sp = sqrt(3593.26/58)
Sp = sqrt(61.9203)
Sp ≈ 7.870
Step 4: Calculate the test statistic (t-value):
t = (x̄1 - x̄2) / (Sp * sqrt(1/n1 + 1/n2))
Where:
x̄1 = mean of treatment group
x̄2 = mean of placebo group
Sp = pooled standard deviation
n1 = size of treatment group
n2 = size of placebo group
Using the given values:
x̄1 = 65.2
x̄2 = 70.3
Sp ≈ 7.870
n1 = 30
n2 = 30
t = (65.2 - 70.3) / (7.870 * sqrt(1/30 + 1/30))
t = -5.1 / (7.870 * sqrt(1/30 + 1/30))
t = -5.1 / (7.870 * sqrt(2/30))
t = -5.1 / (7.870 * sqrt(1/15))
t ≈ -5.1 / (7.870 * 0.2582)
t ≈ -5.1 / 2.0311
t ≈ -2.51
Step 5: Calculate the P-value:
Since we want to find the probability that the difference in means is greater than 0, we need to calculate the P-value for a one-tailed test.
The P-value can be obtained from a t-distribution table or using software. For a t-value of -2.51 with 58 degrees of freedom (n1+n2-2), the P-value is approximately 0.007 (assuming a two-tailed test).
Since we are conducting a one-tailed test, we divide the P-value by 2 to get the final P-value.
P-value = 0.007 / 2
P-value ≈ 0.0035
Step 6: Make a decision:
Compare the P-value to the significance level (alpha) to make a decision.
P-value (0.0035) < alpha (0.05)
Since the P-value is less than the significance level, we reject the null hypothesis.
Step 7: Make a conclusion:
Based on the analysis, there is sufficient evidence to suggest that the difference in mean pulse rates between the treatment group and placebo group is greater than zero.
To find the probability that the difference in the samples (treatment group - placebo group) is greater than 0, we can use the t-distribution and calculate the t-statistic.
Here are the steps to find the probability:
1. Define the null and alternative hypotheses:
- Null hypothesis (H₀): There is no difference in pulse rates between the treatment group and the placebo group (μ₁ - μ₂ = 0).
- Alternative hypothesis (H₁): There is a difference in pulse rates between the treatment group and the placebo group (μ₁ - μ₂ ≠ 0).
2. Calculate the degrees of freedom (df):
- Since both groups have 30 patients, the degrees of freedom are given by df = n₁ + n₂ - 2 = 30 + 30 - 2 = 58.
3. Compute the pooled standard deviation (Sp):
- Sp = √((s₁² * (n₁ - 1) + s₂² * (n₂ - 1)) / (n₁ + n₂ - 2))
- Sp = √((7.8² * (30 - 1) + 8.3² * (30 - 1)) / 58)
- Sp ≈ √((60.84 * 29 + 68.89 * 29) / 58)
- Sp ≈ √(1769.56 + 1997.81) / 58)
- Sp ≈ √3767.37 / 58)
- Sp ≈ √64.942)
- Sp ≈ 8.07
4. Calculate the t-statistic (t):
- t = (x̄₁ - x̄₂) / (Sp * √((1 / n₁) + (1 / n₂)))
- t = (65.2 - 70.3) / (8.07 * √((1 / 30) + (1 / 30)))
- t = -5.1 / (8.07 * √(0.0333 + 0.0333))
- t = -5.1 / (8.07 * √(0.0666))
- t = -5.1 / (8.07 * 0.2582)
- t = -5.1 / 2.097
5. Look up the probability (p-value):
- Using a table or statistical software, find the p-value associated with the t-statistic (t) from Step 4 and the degrees of freedom (df) from Step 2.
- In this case, the p-value will represent the probability of obtaining a difference in pulse rates greater than 0, assuming the null hypothesis is true.
By following these steps and using the appropriate t-distribution table or software, you can find the p-value and determine the probability that the difference in the samples is greater than 0.
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