Part B of question 3 please. Engineers must consider the widths of peoples' heads when designing face masks. Suppose men have head widths that are normally distributed with a mean of 6.125 inches and a standard deviation of1.03 inches.a. If one male is randomly selected, find the probability that his head breadth is less than 6.5 inchesb. If a random sample of 25 males is selected, find the probability that the mean of their head breadths is less than 6.5 inches.c. Is there a difference in the two answers? Explain your answer statistically.Turn in:Sheet with answers for (a), (b), (c) showing work for (a), (b).3. Confidence interval and inference problems (Use formulas given in Chapter 7)Write a confidence interval, showing work, and a statement describing the inference made for each problem:a. For a Statistics course, students are known to have to study daily in order to do well. Data was collected for a sample of 35 stats students from the course and they were found to study for stats for amean of 45 minutes a day. The standard deviation for all the stats students was known to be 5.4. Find a 95% confidence interval for the mean number of minutes that all the stats students study perday.b. A new kind of Covid vaccine is being developed by a medical research team. To test the variability of the new vaccine, a random sample of 16 people were given the new vaccine. Regular blood testsshowed that the sample standard deviation of protection times was 4.7 months. Find a 90% confidence interval for the population standard deviation protection time for the new vaccine.c. A local Dallas basketball legend was known to make a lot of donations to nearby charities. A sample of 30 of his donations had a mean of $41,000 with a standard deviation of $410. Using thissample, find a 99% confidence interval for the mean donation amount for all of his donations.Turn in:Sheet with confidence intervals, including work, and inference statements for the 3 problems. Show work for finding confidence intervals.4. Confidence Interval with Sampling:a. From the list of all 100 Age data in the Data Bank on pages 794-796, select a sample of 30 Age data values, using one of the sampling methods we have studied. Find the proportion (fraction,decimal, or percent) of data in your sample which are greater than 60. Construct a 95%tconfidence interval for the proportion of ages greater than 60 in your sample.b. Count the number of data values in the population (list of all 100 Age data in text) which are greater than 60. Change this number to a proportion (fraction, decimal, or percent) out of the entire list of100.c. Is your proportion, found in (b), between the minimum and maximum values of the 95% confidence interval that you constructed in (a)? Explain why or why not.Turn in:Sheet with (a) the raw data for your sample of 30 Age data; the name of the sampling method and how you conducted the sampling of the 30; the proportion for your sample of 30 data valuesthat are greater than 60; and the confidence interval; (b) the proportion of the population; and (c) answers to questions in (c). Show work for the confidence interval.Class DiscussionClass DiscussionGo to the Discussion Board and participate in the Discussion for this Session.(To get to the Discussion Board simply click on the "Discussion Board" link on the left side of the Screen.)MacBook AirDD80O00F10F11F12F5F6F7F8F9F2F3F4

here sample standard deviation = s= 4.7 sample size n = 16

day. b. A new kind of Covid vaccine is being developed by a medical research team. To test the variability of the new vaccine, a random sample of 16 people were given the new vaccine. Regular blood tests showed that the sample standard deviation of protection times was 4.7 months. Find a 90% confidence interval for the population standard deviation protection time for the new vaccine. A losol Dellee beke