Question

As part of the Mathematics Assessment, eighth-graders were asked about the frequency with which they used calculators while taking tests or quizzes. The results for national public schools were as follows: never, 28%; sometimes, 51%; and always, 21%. A random sample of 140 eighth- grade students in a large urban school district indicated that 30 said never, 78 said sometimes, and 32 said always. At alpha= 0.05, do these proportions differ from the national report?

Work must be shown with steps taken on T 84+ calculator.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: The proportion does not differ from national report.

Alternative hypothesis: The proportion differs from national report.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square goodness of fit test of the null hypothesis.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 3 - 1
D.F = 2
(E_i) = n * p_i

X² = 2.999

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, E_i is the expected frequency count for level i, O_i is the observed frequency count for level i, and X² is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 2.999.

We use the Chi-Square Distribution Calculator to find P(X² > 2.999) = 0.223

Interpret results. Since the P-value (0.223) is greater than the significance level (0.05), we failed to reject the null hypothesis.

At the 0.05 level of significance, we do not have sufficient evidence to conclude that the proportions differ from the national report.

Steps for Ti 84+

1) Click 2nd < Stat < Edit < For L1 - enter observed value and L2 - enter expected value.

2) Click STAT < Tests < X²-GOF test.

3) Enter L1 as Observed, L2 as expected.

4) Enter n -1 as d.f

5) Now click on Calculate.