Answer to Question #99788 in Statistics and Probability for kendall mathieu

Question

In 1990 approximately 77% of young adults in the US felt that the American Dream was achievable. A 2019 survey of 125 young adults in the US resulted in 70% believing that the American Dream is achievable. Does this study provide significant evidence to suggest a drop in the perception of American Dream as achievable?

Answer 1

Suppose the null hypothesis is, $H_0:P=0.77$

and the alternate hypothesis is, $H_a:P<0.77$

These hypotheses constitute a one tailed test. The null hypotheses will be rejected if the sample proportion is too small.

We will use one sample z-test.

$\sigma=\sqrt{P(1-P)/n}$

$=\sqrt{(0.77*0.23)/125}$

$=\sqrt{0.1771/125}$

$=0.0376$

$z=(p-P)/\sigma$

$=(0.70-0.77)/0.0376$

$=-0.07/0.0376$

$=-1.86$

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -1.86

We use Normal distribution table to find $P(z < -1.86) = 0.031$  . Thus, the $P-value = 0.031$

We will check at different significance level.

Case 1: Suppose that the significance level is $\alpha=0.10$

Now since the P-value is less than the significance level, we can reject the null hypothesis and claim with significant evidence that the perception of achieving American dream has dropped.

Case 2: Suppose the significance level is $\alpha=0.05$

Now since the P-value is less than the significance level, we can reject the null hypothesis and claim with significant evidence that the perception of achieving American dream has dropped.

Case 3: Suppose the significance level is $\alpha=0.01$

Now since the P-value is greater than the significance level, we cannot reject the null hypothesis and cannot claim with significant evidence that the perception of achieving American dream has dropped.

Thus, it can be seen that if the significance level will be greater than 0.03, then it can be said with strong evidence that the perception of achieving American dream has dropped.