A survey was carried out to investigate the amount and type of exercise that adults in Ireland are taking in 2022 - Leaving Cert Mathematics - Question 10 - 2022

A disadvantage is that the results may not be as accurate as those obtained from surveying the entire population. Sampling bias can occur if the sample is not representative of the population, leading to skewed results.

To calculate the margin of error (M.E.), we use the formula:

M.E. = 1/eta_{n}

Where eta_{n} = rac{1500}{100}\text{ (as this represents the sample size)}.

Thus: M.E. = rac{1}{ ext{sqrt}(1500)} \approx 0.0258 \times 100 \approx 2.6\%

To find the number of people who walk for recreation, calculate:

$1500 \times \frac{71}{100} = 1065$

Thus, 1065 people from the sample said that they walk for recreation.

Using the percentages from parts (b)(i) and (b)(ii):

• Percentage who walk for recreation: 71%
• Margin of error: 2.6%

The 95% confidence interval can be calculated as:

$[71 - 2.6, 71 + 2.6] = [68.4, 73.6]$

Therefore, the 95% confidence interval is [68.4%, 73.6%].

1. Hypotheses:

   • Null Hypothesis: $H_0 = 65\%$ (The proportion of adults walking for recreation has not changed.)
   • Alternative Hypothesis: $H_a \neq 65\%$ (The proportion of adults walking for recreation has changed.)

2. Test Statistic Calculation:
   Using the sample proportion ($\hat{p} = 0.71$) and assuming the null hypothesis is true:
   $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.71 - 0.65}{\sqrt{\frac{0.65(1-0.65)}{1500}}}$

   • Calculate the above to get the z-value.

3. Compare with Critical Values:
   Use critical z-values for a two-tailed test at the 5% level (z ≈ ±1.96).
   If calculated z is outside of that range, reject the null.

4. Conclusion:
   Conclude whether to reject or not reject $H_0$ based on this z-value.

5. Reason:
   The result shows that the figure of 65% is significantly different from the confidence interval of [68.4%, 73.6%]. Thus, based on the hypothesis test, we conclude that the figure of 65% has changed in 2022.