Question

are you smarter than a second - grader? a random sample of 61 second - graders in a certain school district are given a standardized mathematics skills test. the sample mean score is \\(\bar{x} = 51\\). assume the standard deviation of test scores is \\(\sigma = 10\\). the nationwide average score on this test is 54. the school superintendent wants to know whether the second - graders in her school district have different math skills than the nationwide average. use the \\(\alpha = 0.01\\) level of significance and the \\(p\\) - value method with excel.\
part: 0 / 5\
part 1 of 5\
(a) state the appropriate null and alternate hypotheses.\
\\(h_0:\mu = 54\\)\
\\(h_1:\mu \
eq 54\\)\
this hypothesis test is a two - tailed test.\
part: 1 / 5\
part 2 of 5\
(b) compute the value of the test statistic. round the answer to at least two decimal places.\
\\(z = - 2.34\\)\
part: 2 / 5\
part 3 of 5\
(c) find the \\(p\\) - value. round the answer to at least four decimal places.\
\\(p\\) - value = 0.0193\
part: 3 / 5\
part 4 of 5\
(d) determine whether to reject \\(h_0\\).\
reject the null hypothesis \\(h_0\\).\
part: 4 / 5\
part 5 of 5\
(e) state a conclusion.\
there select enough evidence to conclude that the second - graders in the superintendents school district have different math skills than the nationwide average.

Explanation:

Step1: Recall Hypothesis Testing Logic

We compare the P - value (0.0193) with the significance level \(\alpha = 0.01\). If \(P - value<\alpha\), we reject \(H_0\); otherwise, we do not.

Step2: Compare P - value and \(\alpha\)

Here, \(P - value = 0.0193\) and \(\alpha=0.01\). Since \(0.0193>0.01\), we do not reject \(H_0\). Wait, but in the given part (d) the selected option is "Reject", which seems contradictory. Wait, maybe I made a mistake. Wait, the test statistic \(z=- 2.34\), for a two - tailed test, the P - value is \(2\times P(Z < - 2.34)\). \(P(Z < - 2.34)=0.0096\), so the P - value should be \(2\times0.0096 = 0.0192\approx0.0193\). And \(\alpha = 0.01\). Since \(0.0193>0.01\), we should not reject \(H_0\). But the given part (d) has "Reject" selected, which is incorrect. But maybe the user wants to follow the steps. Wait, for part (e), we need to state the conclusion. If we do not reject \(H_0\), then there is not enough evidence. But let's check again.

Wait, the significance level \(\alpha = 0.01\). The P - value is approximately 0.0193. Since \(P - value>\alpha\), we fail to reject the null hypothesis. So the conclusion is: There is not enough evidence to conclude that the second - graders in the superintendent’s school district have different math skills than the nationwide average.

But let's go through each part:

Part (a)

The null hypothesis \(H_0:\mu = 54\) (the district average is equal to the nationwide average) and the alternative hypothesis \(H_1:\mu
eq54\) (the district average is different from the nationwide average), which is a two - tailed test. This is correct.

Part (b)

The formula for the z - test statistic in a one - sample z - test is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Here, \(\bar{x} = 51\), \(\mu = 54\), \(\sigma = 10\), \(n = 61\). So \(z=\frac{51 - 54}{\frac{10}{\sqrt{61}}}=\frac{- 3}{\frac{10}{7.8102}}\approx\frac{-3}{1.2804}\approx - 2.34\). This is correct.

Part (c)

For a two - tailed test, \(P - value=2\times P(Z < - 2.34)\). Looking up in the standard normal table, \(P(Z < - 2.34)=0.0096\), so \(P - value = 2\times0.0096 = 0.0192\approx0.0193\). This is correct.

Part (d)

Since \(P - value=0.0193>\alpha = 0.01\), we do not reject \(H_0\). But the given option in the image is "Reject", which is an error.

Part (e)

Since we do not reject \(H_0\), there is not enough evidence at the \(\alpha = 0.01\) level of significance to conclude that the second - graders in the superintendent’s school district have different math skills than the nationwide average.

Answer:

Part (a)

\(H_0:\mu = 54\), \(H_1:\mu
eq54\), two - tailed test.

Part (b)

\(z=-2.34\)

Part (c)

\(P - value = 0.0193\)

Part (d)

Do not reject \(H_0\) (Note: The given "Reject" in the image is incorrect as \(0.0193>0.01\))

Part (e)

There is not enough evidence to conclude that the second - graders in the superintendent’s school district have different math skills than the nationwide average.