Question

a random sample of 78 eighth grade students scores on a national mathematics assessment test has a mean score of 277. 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 270. assume that the population standard deviation is 36. at α = 0.08, 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.

a. ( h_0:muleq270 ) (claim)
( h_a:mu > 270 )

b. ( h_0:mugeq270 ) (claim)
( h_a:mu < 270 )

c. ( h_0:muleq270 )
( h_a:mu > 270 ) (claim)

d. ( h_0:mu = 270 )
( h_a:mu > 270 ) (claim)

e. ( h_0:mu = 270 ) (claim)
( h_a:mu > 270 )

f. ( h_0:mu < 270 )
( h_a:mugeq270 ) (claim)

(b) find the standardized test statistic z.
( z=) (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the z - test statistic

The formula for the z - test statistic when the population standard deviation \(\sigma\) is known is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)

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

We are given that \(\bar{x} = 277\), \(\mu=270\), \(\sigma = 36\), and \(n = 78\)

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

First, calculate \(\sqrt{78}\approx8.83\), then \(\frac{36}{8.83}\approx4.08\)

$$ z=\frac{7}{4.08}\approx1.72 $$

Answer:

\(z = 1.72\)