The Company Slimdown Makes Weight Loss Products It Has Recen

The Company Slimdown Makes Weight Loss Products It Has Recently Come

The company Slimdown has introduced a new skin cream claimed to enhance weight loss when applied behind the ears. To evaluate the effectiveness of this product, a randomized sample of 45 individuals was selected, and their percentage weight loss over an eight-week period was recorded. The key objective is to determine the power of the statistical test used to evaluate the cream's efficacy, specifically when the true mean percentage weight reduction is 2.4%. The population standard deviation is known to be 7%, and the significance level for the test is set at α=0.03. Calculating the power of this test involves understanding the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true, particularly at a true mean weight loss of 2.4%.

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Statistical power is a fundamental concept in hypothesis testing when assessing the effectiveness of a treatment or intervention. It quantifies the probability of correctly rejecting the null hypothesis (H₀) when the alternative hypothesis (H₁) is true. In this context, the null hypothesis posits that the mean percentage weight loss (μ) is equal to a certain value, typically the minimum or an expected threshold, while the alternative suggests that the true mean exceeds or differs from this value. Computing the power involves calculating the likelihood that the test statistic will fall into the rejection region under the alternative hypothesis, considering the specified sample size, population standard deviation, significance level, and the actual effect size — in this case, a true mean weight loss of 2.4%.

The given data outline a scenario where a sample of n=45 individuals is used to test the efficacy of a new weight loss cream. The population standard deviation (σ) is known and equals 7%, and the significance level (α) is set at 0.03, indicating a 3% threshold for rejecting the null hypothesis. To compute the power, the first step is to specify the null hypothesis, typically H₀: μ = μ₀, and the alternative hypothesis, which may be H₁: μ > μ₀ if testing for a greater mean weight loss. Although the specific null value (μ₀) isn't explicitly provided, for illustration, assume the null hypothesis is μ₀ = 0% (no weight loss), and the alternative is μ > 0%. Since the true mean weight loss is actually 2.4%, this represents the effect size for the power calculation.

Next, we calculate the standard error of the mean (SE) using the known population standard deviation and the sample size: SE = σ/√n = 7/√45 ≈ 1.043. The critical value (zα) corresponding to the significance level (α=0.03) for a one-tailed test is obtained from standard normal distribution tables. For α=0.03, the critical z-value is approximately 1.88, as the area in the tail is 0.03.

Using the critical z-value, the cutoff point for the test statistic in terms of the sample mean (x̄) is calculated as:

X̄_critical = μ₀ + zα * SE. Assuming μ₀=0%, this yields:

X̄_critical = 0 + 1.88 * 1.043 ≈ 1.96%

This means that if the observed mean weight loss exceeds approximately 1.96%, then the null hypothesis would be rejected at the 3% significance level.

Now, considering the true mean weight loss is 2.4%, we calculate the probability that the sample mean exceeds this cutoff value under the true mean. The standardized z-value for the true mean is calculated as:

z = (X̄ - μ_true) / SE

Substituting the known values:

z = (1.96 - 2.4) / 1.043 ≈ -0.419

Finally, the power is the probability that this Z-score is greater than the negative of zα (since we're dealing with the upper tail in a right-tailed test). Mathematically, this is:

Power = P(Z > (X̄_critical - μ_true) / SE) = 1 - Φ(z)

where Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution at z. Using the calculated z-value of -0.419:

Φ(-0.419) ≈ 0.337

Therefore, the power is approximately:

Power ≈ 1 - 0.337 = 0.663 or 66.3%

In conclusion, based on the given parameters and assumptions, the power of the test to detect a true mean weight loss of 2.4% at a 3% significance level is approximately 66.3%. This indicates a moderate probability of correctly rejecting the null hypothesis when the true effect exists, which is essential for evaluating the efficacy of the new cream in practical settings.

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