A research paper reports, "… in a sample with 350 observations, the sample average Y = 0.9 is statistically significantly different from zero at a 5% level, since its t‐statistic is 3.0." Suppose you want to test the null hypothesis that the true mean of the population is equal to one, not zero. Would you reject this null hypothesis (against a two‐sided alternative) at a 5% level?

I would reject the null hypothesis.

Step-by-step explanation:

The test is a two-tailed test because it is stated that the alternate hypothesis is two-sided.

Degree of freedom = n - 1 = 350 - 1 = 349

At 349 degrees of freedom and 5% significance level, the critical values are -1.96702 and 1.96702

Conclusion:

Reject the null hypothesis because the test statistic 3 falls outside the region bounded by the critical values -1.96702 and 1.96702.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.