Question

1. Height of 9-year olds (Independent Samples Z – p – value)

At age 9, the average weight (28.6 kg) and height (133.3 cm) for both boys and girls are believed to be exactly the same. A random sample of 9 year-olds yielded these results. Is there a statistically significant difference in the heights of boys and girls at age 9?

	Boys	Girls
Sample Size	60	50
Mean height, cm	123.5	126.2
Population variance	98	120

Step 1: State the hypotheses.

Step 2: Compute the test value.

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Step 3: Find the p – value.

Step 4: Make the decision.

Step 5: Summarize the results.

Answer 1

Step 1) We have to test

$\fn_jvn H_0:\mu _1-\mu _2=0\\ H_a:\mu _1-\mu _2\neq 0$

Step 2) The test statistic is

$\fn_jvn z=\frac{\overline{x_1}-\overline{x_2} }{\sqrt{\frac{\sigma _1^2}{n_1}+\frac{\sigma _2^2}{n_2}}}\\$

$\fn_jvn z=\frac{123.5-126.2 }{\sqrt{\frac{98}{60}+\frac{120}{50}}}\\ {\color{Blue} z=-1.3444}$

Since the averages of boys and girls are assumed to be equal $\fn_jvn \mu _1-\mu_2=0$ . Since the population standard deviations are known, we use z-distribution.

Step 3) Here, Here, the sample means are $\fn_jvn \overline{x_1}=123.5 ,\overline{x_2}=126.2$ . The population standard deviations are $\fn_jvn \sigma _1^2=98 ,\sigma _2^2=120$ . The sample sizes are $\fn_jvn n_1=60,n_2=50$ .

The P-value of the test is

$\fn_jvn \textup{p-value}=2P\left ( Z<z \right )\\ \textup{p-value}=2P\left ( Z<-1.3444\right )\\ {\color{Blue} \textup{p-value}=0.1788}$

Step 4) Since $\fn_jvn \textup{p-value}=0.1788>\alpha =0.05$ , we accept the null hypothesis.

Step 5) We conclude that "We are 95% confident that there is no statistically significant difference in the heights of boys and girls at age 9"