In 2015, in a particular country, the weights of 15 year olds were normally distributed with a mean of 63.5 kg and a standard deviation of 10 kg - Leaving Cert Mathematics - Question 8 - 2017

Since 1.5% of 15 year olds are heavier than Kamal, this means that:

$P(Z > z) = 0.015$

Thus, we can find:

$P(Z < z) = 1 - 0.015 = 0.985$

Looking up in the standard normal distribution table, we find:

$z = 2.17$

Next, we can calculate Kamal's weight using the z-score formula:

$z = \frac{X - \mu}{\sigma}$

Rearranging gives:

$X = z \cdot \sigma + \mu$

Substituting the values:

$X = 2.17 \cdot 10 + 63.5 = 85.2 \text{ kg}$

Thus, Kamal’s weight is approximately 85.2 kg.

Null Hypothesis (H₀)

The mean weight of 15 year olds has not changed: $H_0 : \mu = 63.5$

Alternative Hypothesis (H₁)

The mean weight of 15 year olds has changed: $H_1 : \mu \neq 63.5$

To test the hypothesis, we calculate the z-score using the sample statistics:

• Sample mean ($\bar{x}$) = 62 kg
• Sample size (n) = 150
• Standard deviation (s) = 10 kg

Using the z-score formula: $z = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{62 - 63.5}{10 / \sqrt{150}} = -1.8371$

Conclusion

At the 5% significance level, we compare the calculated z-score with critical z-values (-1.96 and 1.96). Since -1.8371 is within -1.96 and 1.96, we fail to reject the null hypothesis. Therefore, there is insufficient evidence to suggest that the mean weight has changed from 2015 to 2016.