Are you smarter than a fifth-grader? A random sample of 60 fifth-graders in a ce
ID: 3311110 • Letter: A
Question
Are you smarter than a fifth-grader? A random sample of 60 fifth-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is = 52. Assume the standard deviation of test scores is = 15. The nationwide average score on this test is 50. The school superintendent wants to know whether the fifth-graders in her school district have greater math skills than the nationwide average. Use a 0.01 level of significance to test the claim that the fifth-graders in her school district have higher math skills than the nationwide average.
State the claim in symbolic form. µ 50 (b) µ 52 (c) µ 50 (d) µ = 50 (e) µ 52
Identify the null and alternative hypotheses: (A) Ho: µ 50 H1: µ 50 (b) Ho: µ = 52 H1: µ 52 (c) Ho: µ 52 H1: µ = 52 (d) Ho: µ 50 H1: µ 50
What type of hypothesis test is this? left-tail (b) right-tailed (c) two-tailed
What is the value of the test statistic? (a) -1.03 (b) 0.8485 (c) 1.03 (d) .1515 (e) 1.63 What is the P-value? (a) 0.8485 (b) 1.03 (c) 1.63 (d) 0.1515 (e) -1.03
State the final conclusion in simple non-technical terms. Be sure to address the original claim (hint: see figure 8-7 on page 403).
A.There is not sufficient sample evidence to support the claim that the fifth-graders have higher math skills than the nationwide average.
B.There is sufficient sample evidence to warrant rejection of the claim that the fifth-graders have higher math skills than the nationwide average.
C.There is not sufficient evidence to warrant rejection of the claim that the fifth-graders have higher math skills than the nationwide average.
D.The sample data support the claim that the fifth-graders have higher math skills than the nationwide average.
Explanation / Answer
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: <= 50
Alternative hypothesis: > 50 (Claim)
Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too large.
Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a one-sample t-test.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = s / sqrt(n) = 15 / sqrt(60) = 1.9365
DF = n - 1 = 60 - 1 = 59
t = (x - ) / SE = (52 - 50)/1.9365 = 1.03279
where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.
Here is the logic of the analysis: Given the alternative hypothesis ( > 50), we want to know whether the observed sample mean is large enough to cause us to reject the null hypothesis.
The observed sample mean produced a t statistic test statistic of 1.03279. We use the t Distribution Calculator to find P(t < 1.03279)
The P-Value is 0.152956.
The result is not significant at p < 0.01.
Interpret results. Since the P-value (0.152956) is greater than the significance level (0.01), we cannot reject the null hypothesis.
There is not sufficient sample evidence to support the claim that the fifth-graders have higher math skills than the nationwide average.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.