Answer :
a)A 90% confidence interval is constructed to estimate the mean difference in pain ratings between the control and experimental groups. the confidence interval is (−5.399, 15.199).b)the true mean difference lies within this interval, indicating that the experimental treatment may have lower or similar pain ratings compared to the old treatment.
The hypothesis test is then performed to determine if there is sufficient evidence to show that the experimental treatment has lower pain ratings on average than the old treatment, using a significance level of α = 0.10.
a. To construct the 90% confidence interval, we need to calculate the standard error and the margin of error. The standard error can be calculated as the square root of [(σ₁²/n₁) + (σ₂²/n₂)], where σ₁² and σ₂² are the variances of the control and experimental groups respectively, and n₁ and n₂ are the sample sizes. Substituting the given values, the standard error is approximately 5.773. The margin of error is then obtained by multiplying the standard error by the critical value for a 90% confidence level, which is 1.645. The margin of error is approximately 9.499. Finally, we construct the confidence interval by subtracting the margin of error from the sample mean difference and adding the margin of error to the sample mean difference. In this case, the confidence interval is (−5.399, 15.199).
b. The 90% confidence interval (−5.399, 15.199) suggests that, based on the sample data, we are 90% confident that the true mean difference in pain ratings between the control and experimental groups falls within this interval. This means that we cannot conclude with certainty that the experimental treatment has lower pain ratings on average than the old treatment, as the interval includes zero (no difference). However, it is possible that the true mean difference lies within this interval, indicating that the experimental treatment may have lower or similar pain ratings compared to the old treatment. Further analysis or a larger sample size may be needed to make a more definitive conclusion.
For the hypothesis test:
Null hypothesis (H₀): The experimental treatment does not have lower pain ratings on average than the old treatment.
Alternative hypothesis (H₁): The experimental treatment has lower pain ratings on average than the old treatment.
The significance level (α) is given as 0.10.
Test statistic: We can use a t-test to compare the means of two independent groups. The test statistic can be calculated as (Y₁ - Y₂) / sqrt((σ₁²/n₁) + (σ₂²/n₂)). Substituting the given values, the test statistic is approximately 0.706.
p-value calculation: Using the test statistic, the degrees of freedom (df) can be calculated as (n₁ + n₂ - 2), which is 48 in this case. Consulting a t-distribution table or using statistical software, we find that the p-value is approximately 0.483.
Since the p-value (0.483) is greater than the significance level (0.10), we fail to reject the null hypothesis. This means that we do not have sufficient evidence to show that the experimental treatment has lower pain ratings on average than the old treatment at the α = 0.10 level. However, it's important to note that this conclusion may change with different significance levels or larger sample sizes.
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a. The 90% confidence interval to estimate the mean difference in pain ratings between the control and experimental groups is (-0.3, 7.1).
b. This means that with 90% confidence, the true mean difference in pain ratings between the control and experimental groups is likely to be between -0.3 and 7.1 units.
a. To construct the 90% confidence interval for the mean difference in pain ratings, we can use the formula:
CI = (Y1 - Y2) ± t * √[tex]((s1^2/n1) + (s2^2/n2))[/tex]
where Y1 and Y2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and t is the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom.
Plugging in the given values, we have:
CI = (78.1 - 74.1) ± t * √((51.7^2/25) + (51.7^2/25))
CI ≈ 4 ± 1.714 * √(10.6688 + 10.6688)
CI ≈ 4 ± 1.714 * √(21.3376)
CI ≈ 4 ± 1.714 * 4.6187
CI ≈ (4 - 7.9207, 4 + 7.9207)
CI ≈ (-0.3, 7.1)
b. This 90% confidence interval indicates that, with 90% confidence, the true mean difference in pain ratings between the control and experimental groups is likely to fall between -0.3 and 7.1 units. The interval contains both positive and negative values, suggesting that the experimental group's mean pain rating could be either lower or higher than the control group's mean. However, since the interval includes zero (no difference), we cannot definitively conclude that there is a significant difference in mean pain ratings between the two groups based on this confidence interval alone.
To perform a hypothesis test to determine if the experimental treatment has lower pain ratings on average than the old treatment, we need to follow the 5-step process of hypothesis testing. Please provide the necessary information for the hypothesis test, such as the null and alternative hypotheses and the significance level.
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