A political poll estimates that 48% of Democrat voters will vote for Elizabeth Warren in the presidential primaries. They provid
e a "margin of error" of 3 percentage points, which leads to a 95% confidence interval of 45% to 51%. a) Explain what a 95% confidence level means. b) why is a confidence interval needed here? What does it tell us that the "48%" does not tell us? c) Which of the following is the best interpretation of the confidence interval? i) 95% of the samples had between 45% and 51% of respondents supporting Warren. ii) There's a 95% chance that between 45% and 51% of those sampled support Warren. iii) We are 95% confident that between 45% And 51% of all voters support Warren. iv) We are 95% confident that between 45% and 51% of those sampled support Warren. 3. Educators introduce a new math curriculum and measure 45 students' scores before and at the end of the academic year using an exam widely believed to accurately measure students understanding. The average score at the start the year was 65%, and after was 75%. a) In words, what are the null and alternative hypothesis tests that the educators would most likely be interested in testing? b) What does a hypothesis test tell us that we can't learn from simply noting that students improved 10 percentage points? c) Suppose the p-value from this hypothesis test was 0.54. Explain how to interpret this value in this context.
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for all
in [-3, 0], so
is non-decreasing over this interval, and in particular we know right away that its minimum value must occur at
.
From the plot, it's clear that on [-3, 0] we have . So
for some constant . Given that
, we find that
so that on [-3, 0] we have
and