From the uploaded image, I can help guide you through solving the problem step-by-step.

Step-by-Step Solution:

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$:

• The problem mentions testing if the proportion of gamers who experience motion sickness from using new VR glasses is more than 34%.
• Null Hypothesis ($H_0$): $p = 0.34$
• Alternative Hypothesis ($H_1$): $p > 0.34$ (This is a one-tailed test because we are testing "more than 34%").

(b) Calculate $np$ and $n(1-p)$:

• Given:
  • Sample size $n = 195$
  • Hypothesized proportion $p = 0.34$
• $np = 195 \times 0.34 = 66.3$
• $n(1-p) = 195 \times (1-0.34) = 195 \times 0.66 = 128.7$

Both $np \geq 10$ and $n(1-p) \geq 10$, so the conditions for using a z-test are satisfied.

(c) Perform the z-test and calculate the p-value:

The formula for the z-test statistic is: $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

• Observed proportion $\hat{p} = \frac{x}{n} = \frac{77}{195} = 0.3949$
• Substitute into the formula:
  • Standard error $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.34 \cdot 0.66}{195}} = \sqrt{\frac{0.2244}{195}} = \sqrt{0.001151} \approx 0.0339$
  • $z = \frac{0.3949 - 0.34}{0.0339} = \frac{0.0549}{0.0339} \approx 1.62$

Determine the p-value:

• Use a z-table or software to find the p-value for $z = 1.62$.
• The p-value corresponds to the area to the right of $z = 1.62$. From z-tables, $P(z > 1.62) \approx 0.0526$.

Conclusion:

• If the significance level $\alpha = 0.05$, the p-value $0.0526 > 0.05$, so we fail to reject the null hypothesis. There is not enough evidence to conclude that the proportion of gamers who experience motion sickness is more than 34%.

If you'd like, I can provide more details, or help with verifying calculations further.

5 Follow-Up Questions:

1. What does a p-value represent in hypothesis testing?
2. How do you decide whether to reject the null hypothesis in a one-tailed test?
3. Why do we check the conditions $np \geq 10$ and $n(1-p) \geq 10$ for a z-test?
4. How would the conclusion change if the significance level were $\alpha = 0.10$?
5. What alternative methods can be used if the sample size is too small for a z-test?

Tip:

Always visualize the hypothesis test on a normal distribution curve to better understand the rejection region and the calculated z-value.