Questions: Researchers studying the effects of diet on growth would like to know if a vegetarian diet affects the height of a child. The researchers randomly selected 14 vegetarian children that were six years old. The average height of the children is 43.5 inches with a standard deviation of 3.4 inches. The average height for all six-year-old children is 45.75 inches. Using confidence intervals, test to determine whether there is overwhelming evidence at α=0.01 that six-year-old vegetarian children are not the same height as other six-year-old children. Assume the population is normally distributed.

Solution Steps

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.4}{\sqrt{14}} \approx 0.9087 \]

The test statistic \( t_{\text{test}} \) is calculated using the formula: \[ t_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{43.5 - 45.75}{0.9087} \approx -2.4761 \]

For a two-tailed test, the P-value is calculated as: \[ P = 2 \times (1 - T(|z|)) \approx 0.0278 \]

Since the P-value \( 0.0278 \) is greater than the significance level \( \alpha = 0.01 \), we conclude that: Because the hypothesized value falls in the confidence interval, we fail to reject the null hypothesis. There is not sufficient evidence at the \( 0.01 \) significance level that six-year-old vegetarian children are not the same height as other six-year-old children.

Final Answer