(Solved): Q1) The mean typing speed in words per minute (WPM) for professional programmers is believed to ...

Q1) The mean typing speed in words per minute (WPM) for professional programmers is believed to be 55 WPM. A sample ( $$n=120$$ ) of programmers from a number of large tech company decide to measure their typing speed. After the 120 programmers have tested their typing speed, the mean typing speed of the sample was 57 WPM with a sample standard deviation of 10 WPM. The group of programmers would like to test the hypotheses that the typing speed for programmers is actually higher than the previously believed 55 WPM. (i) State hypotheses and define appropriate symbols (ii) State what checks and assumption the programmers would have had to make. (iii) Calculate the test statistic for this hypothesis. (iv) What is the degrees of freedom and $$p$$-value for this test (v) One of the people analysing the data makes the statement that "at the $$?=0.01$$ level, we can reject the null hypothesis, and conclude there is sufficient evidence to accept the alternate hypothesis that the mean typing speed of professional programmers is higher than 55 WPM". Is this statement correct? If it is not correct, what type of error has been made and why? (vi) The typing speed of more professional programmers is collected, such that the sample is now $$n=200$$ programmers, with a sample mean of 56.8 and a sample standard deviation of 10.5 . Calculate the new test statistic, degrees of freedom, and $$p$$-value. (vii) One person makes the following statement, "at the $$?=0.01$$ level, we cannot reject the null hypothesis that the typing speed of professional programmers is 55 WPM". Is this statement correct? If it is not correct, what type of error has been made and why? (viii) Another person makes the following statement, "at the $$?=0.01$$ level, we have sufficient evidence to reject the null hypothesis that the population mean typing speed of professional programmers is 55 WPM, and accept the alternate hypothesis that the population mean typing speed is greater than 55 WPM". Is this statement correct? If it is not correct, what type of error. has been made and why?