A 2003 study of dreaming found that out of a random sample of 113 people, 35 reported dreaming in color. However, the proportion reported dreaming in color that was established in the 1940s was 0.25. the researcher wants to determine if the proportion of individuals that dream in color has changed since the 1940s. use 0.10 level of significance.1. What is the normal approximation method appropriate for this test?2. compute appropriate test statistic.3. At a 0.10 level of significance, what are the critical values for the test.4. what Is the appropriate decision and conclude for the test at 0.10 level of significance (fail to reject, reject Ha)5. would your conclusion change if the test were to be conducted as an upper tailed test? Why or why not supporting your answer using the p-value approach.

Answer:1. The appropriate method to test this hypothesis is the z-statistic for a proportion because the distribution of the population is the following:[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]2. [tex]z=\frac{0.310 -0.25}{\sqrt{\frac{0.25(1-0.25)}{113}}}=1.473[/tex]  3. [tex]z_{\alpha/2}=1.64[/tex] [tex]z_{1-\alpha/2}=-1.64[/tex]4. [tex]p_v =2*P(z>1.473)=0.141[/tex]  We fail to reject the null hypothesis is since our statistic is less than the critical values, so we are on a zone of non rejection of the null hypothesis. 5. The conclusion change. (see below)Step-by-step explanation:Data given and notation n  n=113 represent the random sample takenX=35 represent the people reported with dreaming in color[tex]\hat p=\frac{35}{113}=0.310[/tex] estimated proportion of people reported with dreaming in color[tex]p_o=0.25[/tex] is the value that we want to test[tex]\alpha=0.10[/tex] represent the significance levelConfidence=90% or 0.90z would represent the statistic (variable of interest)[tex]p_v[/tex] represent the p value (variable of interest)  1) What is the normal approximation method appropriate for this test?The appropriate method to test this hypothesis is the z-statistic for a proportion because the distribution of the population is the following:[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]We need to conduct a hypothesis in order to test the claim that the proportion of individuals that dream in color has changed since the 1940s:  Null hypothesis:[tex]p=0.25[/tex]  Alternative hypothesis:[tex]p \neq 0.25[/tex]  When we conduct a proportion test we need to use the z statistic, and the is given by:  [tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].2) compute appropriate test statistic.Since we have all the info requires we can replace in formula (1) like this:  [tex]z=\frac{0.310 -0.25}{\sqrt{\frac{0.25(1-0.25)}{113}}}=1.473[/tex]  3) At a 0.10 level of significance, what are the critical values for the test.For this case the confidence is 90% or 0.9. We can calculate [tex]\alpha=1-0.9=0.1[/tex] and [tex]\alpha/2=0.05[/tex], we need two critical values since we have a bilateral test, we need two values on the normal standard distribution such that:[tex]P(Z>a)=0.05 ,P(Z<-a)=0.05[/tex]Ans using the following codes in Excel "=NORM.INV(0.05,0,1)" and "=NORM.INV(1-0.05,0,1)" we got that the critical values are:[tex]z_{\alpha/2}=1.64[/tex] [tex]z_{1-\alpha/2}=-1.64[/tex]4) what Is the appropriate decision and conclude for the test at 0.10 level of significance (fail to reject, reject Ha)It's important to refresh the p value method or p value approach . "This methos is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  The significance level provided [tex]\alpha=0.1[/tex]. The next step would be calculate the p value for this test.  Since is a bilateral test the p value would be:  [tex]p_v =2*P(z>1.473)=0.141[/tex]  If we compare the p value obtained value and the significance level given [tex]\alpha=0.1[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 10% of significance, that the proportion of dreaming in color changed from 1940s .  The other way to see that we fail to reject the null hypothesis is since our statistic is less than the critical values, so we are on a zone of non rejection of the null hypothesis. 5. would your conclusion change if the test were to be conducted as an upper tailed test? Why or why not supporting your answer using the p-value approach.Null hypothesis:[tex]p\leq 0.25[/tex]  Alternative hypothesis:[tex]p > 0.25[/tex]  On this case the critical value changes since we need a value such that:[tex]P(Z>a)=0.1[/tex]And the critical value is [tex]z_{\alpha}=1.28[/tex]And since our calculated value is higher than the critical value we reject the null hypothesis at 10% of significance.And calculating the p value we got:[tex]p_v =P(z>1.473)=0.070[/tex]  If we compare the p value obtained value and the significance level given [tex]\alpha=0.1[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance, that the proportion of dreaming in coolor is significantly higher from the proportion in 1940s .