Question

A weight loss center provided a loss for 72% of its participants. The center's doctor believes this is not a sufficient percentage of people losing weight. She decides to test a new weight loss program on a random sample size of 140 subjects and found weight loss in 78% of the participants. Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.

Answer 1

Part A;

The null hypothesis, H₀, is that the weight loss obtained in the new program is less than or equal to the weight loss obtained in the original program

The alternative hypothesis, Hₐ, is that the weight loss obtained in the new program is more than the weight loss obtained in the original program

Part B;

There is not evidence to suggest that the new program results in an improvement in the percentage of its participants that had weight loss

Explanation:

Part A

The participants of the weight loss center that, p₀ = 72% = 0.72

The belief of the center's doctor is that the percentage is not sufficient for the proportion of people losing weight

The number of subjects in the sample the center's doctor used for the test, n = 140

The weight loss percent of the sample using the new program,
`\hat p` = 78% = 0.78

The null hypothesis, H₀;
`\hat p` ≤ p₀

The alternative hypothesis, Hₐ;
`\hat p` > p₀

Part B

The z-test is given as follows;

`z=\frac{\hat{p}-p_0}{\sqrt{(p_0 \cdot (1 - p_0))/(n)}}`

`\therefore z=\frac{0.78-0.72}{\sqrt{(0.72 \cdot (1 - 0.72))/(140)}} = 1.53113883009 \approx 1.531`

The p-value = P(Z ≥ 1.531) = 1 - P(Z ≤ 1.531) = 1 - 0.93699 = 0.06301

Given that the p-value = 0.06301 is larger than the significance level of 0.05, we fail to reject the null

Therefore, there is not enough evidence to suggest that the mean of the weight loss of participants in the new method is more than the weight loss percentage obtained from the original weight loss program.