Problem 14 A nutritionist claims that the p... [FREE SOLUTION] (2024)

Short Answer

proportion comparison

Comparing proportions is a key aspect of hypothesis testing. In this exercise, we compared the proportions of females and males who consume too much saturated fat.
Proportion is essentially a fraction or percentage representing part of a whole.
For females, the sample proportion is: \[ p_f = \frac{300}{513} \approx 0.5851 \] (where 300 represents females who consume too much saturated fat out of 513 surveyed).
For males, the sample proportion is: \[ p_m = \frac{391}{564} \approx 0.6936 \] (where 391 males consume too much saturated fat out of 564 surveyed).
The calculation of these proportions is fundamental to determine if there's a significant difference between the two sexes when it comes to saturated fat consumption.

confidence interval

A confidence interval gives us an estimated range of values which is likely to include the difference between population proportions.
In our example, we need a 95% confidence interval for the difference between two proportions.
The difference in sample proportions is: \[ p_f - p_m = 0.5851 - 0.6936 = -0.1085 \]
The margin of error is calculated as: \[ E = Z_{0.025} * SE = 1.96 * 0.0285 \approx 0.0559 \]
Thus, the 95% confidence interval for the difference is: \[ (-0.1085 - 0.0559, -0.1085 + 0.0559) = (-0.1644, -0.0526) \]
Since the interval is entirely negative, it indicates that the true proportion of females who consume too much saturated fat is less than that of males.

significance level

The significance level, denoted by \( \alpha \), is a threshold we set to determine if an observed effect is statistically significant.
In this exercise, the significance level is set at 0.05, which means there is a 5% risk of concluding that a difference exists when there is no actual difference.
The critical value for a one-tailed test with \( \alpha = 0.05 \) is -1.645.
The test statistic computed is: \[ Z = \frac{p_f - p_m}{SE} = \frac{0.5851 - 0.6936}{0.0285} \approx -3.81 \]
Since \( -3.81 < -1.645 \), we reject the null hypothesis.
This means we have enough evidence to support the nutritionist's claim at the 0.05 significance level.