2.2 Shilunga’s weight loss centre provided a loss for 72% of its participants. Shilunga
decides to test a new weight loss programme on a random sample of 140 participants
and found that 91 participants lost weight. Should the centre continue using its new
programme? Test an appropriate hypothesis using 1% level of significance. [8]

To determine whether Shilunga's weight loss center should continue using the new weight loss program, we'll perform a hypothesis test using a 1% level of significance. This test will help us determine if the proportion of participants who lost weight under the new program is significantly different from the previous proportion of 72%.

### Step 1: State the Hypotheses
- **Null Hypothesis (H₀):** \( p = 0.72 \)  
  (The proportion of participants who lost weight with the new program is the same as the previous program.)
- **Alternative Hypothesis (H₁):** \( p \neq 0.72 \)  
  (The proportion of participants who lost weight with the new program is different from the previous program.)

### Step 2: Set the Level of Significance
- The level of significance (\( \alpha \)) is 0.01 (1%).

### Step 3: Calculate the Test Statistic
We will use a **z-test** for the proportion since the sample size is large.

Given:
- \( p_0 = 0.72 \) (previous proportion)
- \( \hat{p} = \frac{91}{140} = 0.65 \) (sample proportion under the new program)
- \( n = 140 \) (sample size)

The formula for the test statistic \( z \) is:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

Substitute the given values:

\[
z = \frac{0.65 - 0.72}{\sqrt{\frac{0.72(1 - 0.72)}{140}}}
\]

### Step 4: Calculate the Critical Value and p-value
We compare the calculated z-value with the critical z-value for a two-tailed test at \( \alpha = 0.01 \).

The critical z-value for a 1% significance level (two-tailed) is approximately \( \pm 2.576 \).

### Step 5: Decision Rule
- If \( |z| > 2.576 \), reject the null hypothesis.
- If \( |z| \leq 2.576 \), do not reject the null hypothesis.

### Step 6: Conclusion

Let's calculate the test statistic to reach a conclusion.

I'll perform the calculations now.The calculated \( z \)-value is approximately \(-1.845\).

### Step 7: Compare the z-value with the Critical Value
- The critical z-value for a two-tailed test at the 1% significance level is \( \pm 2.576 \).
- Our calculated z-value (\( -1.845 \)) does not exceed the critical value in magnitude.

### Conclusion
Since \( |z| = 1.845 \) is less than the critical value of \( 2.576 \), we **fail to reject the null hypothesis**. This means that there is not enough evidence to conclude that the new program is significantly different from the previous one at the 1% significance level.

**Recommendation**: The center should **not discontinue** the new program based on this statistical test, as the results do not show a significant difference in performance compared to the previous program.

---

Do you want details on the z-test process or have any questions?

Here are some related questions to consider:
1. What assumptions are made when performing a z-test for proportions?
2. How would the conclusion change if the significance level was 5% instead of 1%?
3. What is the power of the test, and why is it important in hypothesis testing?
4. How would a different sample size affect the test result?
5. What other factors should Shilunga consider besides the statistical significance?

**Tip:** Always ensure your sample size is large enough when applying a z-test for proportions to achieve reliable results.

Explore the hypothesis testing case study involving Shilunga's weight loss center, determining whether the new weight loss program should be continued based on statistical analysis. Includes step-by-step calculations, significance level determination, and conclusion interpretation.

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