Question

A magazine article states that the mean weight of one-year-old boys is the same as that of one-year-old girls. Does the confidence interval contradict this statement? The confidence interval this statement

Answer 1

Answer:

Yes, the confidence interval contradict this statement.

Step-by-step explanation:

The complete question is attached below.

The data provided is:

$n_{1}=318\\n_{2}=297\\\bar x_{1}=25\\\bar x_{2}=24.1\\s_{1}=3.6\\s_{2}=3.8$

Since the population standard deviations are not provided, we will use the t-confidence interval,

$CI=(\bar x_{1}-\bar x_{2})\pm t_{\alpha/2, (n_{1}+n_{2}-2)}\cdot s_{p}\cdot\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}$

Compute the pooled standard deviation as follows:

$s_{p}=\sqrt{\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}=\sqrt{\frac{(318-1)(3.6)^{2}+(297-1)(3.8)^{2}}{318+297-2}}=2.9723$

The critical value is:

$t_{\alpha/2, (n_{1}+n_{2}-2)}=t_{0.05/2, (318+297-2)}=t_{0.025, 613}=1.962$

*Use a t-table.

The 95% confidence interval is:

$CI=(\bar x_{1}-\bar x_{2})\pm t_{\alpha/2, (n_{1}+n_{2}-2)}\cdot s_{p}\cdot\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}$

     $=(25-24.1)\pm 1.962\times 2.9723\times \sqrt{\frac{1}{318}+\frac{1}{297}}\\\\=0.90\pm 0.471\\\\=(0.429, 1.371)\\\\\approx (0.43, 1.37)$

The 95% confidence interval for the difference between the mean weights is (0.43, 1.37).

To test the magazine's claim the hypothesis can be defined as follows:

H₀: There is no difference between the mean weight of 1-year old boys and girls, i.e. $\mu_{1}-\mu_{2}=0$.

Hₐ: There is a significant difference between the mean weight of 1-year old boys and girls, i.e. $\mu_{1}-\mu_{2}\neq 0$.

Decision rule:

If the confidence interval does not consists of the null value, i.e. 0, the null hypothesis will be rejected.

The 95% confidence interval for the difference between the mean weights does not consists the value 0.

Thus, the null hypothesis will be rejected.

Conclusion:

There is a significant difference between the mean weight of 1-year old boys and 1-year old girls.