Refer to the previous exercise and perform the following tasks.a) Predict for the Director of the Department of Education in Queensland the mean mathematics test score at a school in the state where 50% of the teachers have at least one university degree in mathematics, their mean age is 43 years, and their mean annual income is $78,500. What is the corresponding approximate 95% prediction interval? Calculate the point prediction first manually and then with R, and the approximate 95% prediction interval manually only.b) Predict for the Director of the Department of Education in Queensland the mean mathematics test score at all schools in the state where 50% of the teachers have at least one university degree in mathematics, their mean age is 43 years, and their mean annual income is $78,500.c) Predict with 95% confidence the mean mathematics test score at a school in the state where 50% of the teachers have at least one university degree in mathematics, their mean age is 43 years, and their mean annual income is $78,500. How does this prediction interval compare to the approximate 95% prediction interval you obtained in part (a)? d) Estimate with 95% confidence the mean mathematics test score at schools in the state where 50% of the teachers have at least one university degree in mathematics, their mean age is 43 years, and their mean annual income is $78,500. How does this confidence interval compare to the prediction interval you obtained in part (c)?

predict(m,new_data,se.fit=TRUE, level=0.95,interval="confidence") 截屏2024-05-05 22.49.36.png This indicates that the average mathematics test score, with a 95% confidence interval, is between 57.327 and 80.759 across all schools in the state where 50% of the instructors have at least one mathematics university degree, their mean age is 43 years, and their mean yearly income is $78,500.predict(m,new_data,se.fit=TRUE, level=0.95,interval="confidence") 截屏2024-05-05 22.49.36.png This indicates that the average mathematics test score, with a 95% confidence interval, is between 57.327 and 80.759 across all schools in the state where 50% of the instructors have at least one mathematics university degree, their mean age is 43 years, and their mean yearly income is $78,500.

Predict for the Director of the Department of Education in Queensland the mean mathematics test score at all schools in the state where 50% of the teachers have at least one university degree in mathematics, their mean age is 43 years, and their mean annual income is $78,500.

The Director of the Department of Education in Queensland was analysing the last year average mathematics test scores in the schools under his control. He noticed that there were dramatic differences in scores among the schools. In an attempt to improve the scores of all the schools, he attempted to determine the factors that account for the differences. Accordingly, he took a random sample of 40 schools across the state and, for each, determined the mean mathematics test score, the percentage of teachers in each school who have at least one university degree in mathematics (math), the mean age, and the mean annual income ($ ‘000) of the mathematics teachers. These data are saved in the t10e5 Excel file.a) Perform a multiple regression analysis on these data with R. What is your sample regression equation?b) Is the model useful in explaining the variation among schools? Explain.c) Are the normality and homoskedasticity conditions satisfied? Explain.d) Is multicollinearity a problem? Explain.e) Interpret and the coefficients. Do you find their signs reasonable? Why or why not? Based on your expectations perform t-tests on the coefficients (use a = 0.05).f) Test the null hypothesis that neither the teachers’ mean age nor their mean annual income has a significant effect on the average mathematics test scores (use a = 0.05).

Question 2381 random elementary schools were asked for their average exam scores (sample mean = 535, sample standard deviation = 7). Calculate the 98% confidence interval.1 point(528.00, 542.00)(533.15, 536.85)Not possible to calculate based on this information.(533.41, 536.59)

The percentage y of mathematicians attending the annual conference “Math-ematics in Economics” can be modelled, as a function of the years t, byy = 50 − A e−at.Given that the percentage of participant mathematicians was 2 at the openingyear, and 5 after 2 years,(a) find the value of A and a, correct to 3 decimal places,(b) predict the percentage of mathematicians after 5 years, and(c) predict the percentage of mathematicians attending the conference inthe long run.