Question

a random sample of 75 eighth grade students scores on a national mathematics assessment test has a mean score of 283. this test result prompts a state school administrator to declare that the mean score for the states eighth graders on this exam is more than 280. assume that the population standard deviation is 39. at \\( \alpha = 0.03 \\), is there enough evidence to support the administrators claim? complete parts (a) through (e).
(a) write the claim mathematically and identify \\( h _ { 0 } \\) and \\( h _ { a } \\). choose the correct answer below.
\\( \bigcirc \\) a. \\( h _ { 0 } : \mu \leq 280 \\) (claim) \\( h _ { a } : \mu > 280 \\)
\\( \bigcirc \\) b. \\( h _ { 0 } : \mu \geq 280 \\) (claim) \\( h _ { a } : \mu < 280 \\)
\\( \bigcirc \\) c. \\( h _ { 0 } : \mu < 280 \\) \\( h _ { a } : \mu \geq 280 \\) (claim)
\\( \bigcirc \\) d. \\( h _ { 0 } : \mu = 280 \\) (claim) \\( h _ { a } : \mu > 280 \\)
\\( \bigcirc \\) e. \\( h _ { 0 } : \mu \leq 280 \\) \\( h _ { a } : \mu > 280 \\) (claim)
\\( \bigcirc \\) f. \\( h _ { 0 } : \mu = 280 \\) \\( h _ { a } : \mu > 280 \\) (claim)
(b) find the standardized test statistic z.
\\( z = \square \\) (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the z - statistic

The formula for the z - statistic in a one - sample z - test is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean under the null hypothesis, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Identify the values of \(\bar{x}\), \(\mu\), \(\sigma\), and \(n\)

We are given that \(\bar{x} = 283\), \(\mu=280\) (from the null hypothesis \(H_0:\mu\leq280\)), \(\sigma = 39\), and \(n = 75\).

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

Since \(\sqrt{75}\approx8.66\), then \(3\sqrt{75}\approx3\times8.66 = 25.98\)

$$ z=\frac{25.98}{39}\approx0.67 $$

Answer:

\(z = 0.67\)