Question 1 West Indian manatees travel at a speed that is approximately normally distributed with a mean of 3.8 miles per hour and a standard deviation of 1.34 miles per hour . (a) (2 points ) What proportion of West Indian manatees travel at speeds between 3 and 4.7 miles per hour ? (b) (1 point ) How slow do the slowest 45% of West Indian manatees travel '2' (c) (1 point ) You sample 50 West Indian manatees . What is the sampling distribution of their average speed, X? ((1) (2 points ) What is the probability that the mean speed of your sample is between 3 and 4.7 miles per hour? Question 2 Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy ) There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glu- cose level is above 140 milligrams per deciliter (mg /dl) one hour after having a sugary drink . Shelia's measured glucose level one hour after the sugary drink varies according to the Normal distribution with p: = 122 mg [CH and a: 12mg /dl. (a) (1 point ) If a single glucose measurement is made , what is the probability that Shelia is diagnosed as having gestational diabetes ? (b) (2 points ) If measurements are made on 4 separate days and the mean result is compared with the criterion 140 mg / d1, what is the probability that Shelia is diagnosed as having gestational diabetes? Question 3 (Each part [1 point ]) The 2010 General Social Survey asked the question : "For how many days during the past 30 days was your mental health , which includes stress , depression, and problems with emotions , not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010 A (a) Interpret this interval in context of the data. (b) What does "95% confident" mean ? Explain in the context of the application. (c) Suppose the researchers think a 99% con?dence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% con?dence interval? (d) If a new survey were to be done with 500 Americans ,would the standard error of the esti- mate be larger , smaller , or about the same . Assume the standard deviation has remained constant since 2010.

SOLUTION

Step-by-step explanation

Q1

Mean, p: 3.3 Standard deviation, 0 = 1.34 Z=(X-uilo A. X1 = 3 21 = {3-3.3},(134 = -0.59701 x2 = 4.7 22 = {4.7-3.3}I1.34 = 0.611542 Plant-=43) Can be found from normal table as, P(3<x<4.7] = 0.473345 = 47.38% P(x<Z): 0.45 2 score for which Find] = 0.45 can be found from normal table as, Z = 4.1.126 X = 20+" (—0.126 * 1.34] +3.8: 3.631514 = 3.632 Poplation mean, I1 = 3.8 Population standard deviation, o = 1.34 Sample size, n = 50 For the sampling distribution of sample means, Mean, u, = |-1 = 3.8 Standard deviation, 0,, = :1an = 1.34/1150 = 0.189505 Poplation mean, u = 3.8 Population standard deviation,o = 1.34 Sample size, n = 50 For the sampling distribution of sample means, Mean, pi = p = 3.8 Standard deviation, 0;, = Olin = 1.34.1950 = 0.189505 x1 = 3 21 = [3-3.8]!0.1895 = -4.22153 x2 = 4.7 7.2 = (4.7-3.8)!0.1895 = 4.749225 Pl3<x<4.7}i Can be found from normal table as, P{3<x<4.7} = 0.999987 = 100.00%

Q2 (a)

Answer : Fact I-coare Interested in glucose levels Date : 12 / 1/ 21 Pact 2 - It turns out that sheilas glucose levels very and the distribution is nomad 1: 122 moldl and : 12 mylal pact 3 - A person with glucose levels above 140 maldl is classified as having gestational diabeties. P (x > 140 ) . p ( 2+ 140-122) 12 - P ( 2 > 1. 5 ) = 1 - P(2415) 3 1- 0. 93 32 = 0. 0668 The result suggests is that if we take one reading at random there is a 6.68% chance of having a glucose level reading higher than 140 molds 

(b)

Four measurements as a random sample with common N(122,12) distribution, that means mean (µ) = 122 and σ = 12, therefore sample mean = 122.

Q3

a) Interpret this intreval in the context of the data

>>> we are 95% confident that your mental health is not good is belong to lower limit of confidence limit =3.40 days and upper limit of confidence limit =4.24 days

b) What does 95% confident mean ? Explain in the context of the application.

>>> See 95% confident means (gareanty ) we True parameter value lies between this interval

and 5 % erroer chance .

and   See 95% confident means we mental heath is not this interval of days .

a) We are 95% confident that the true population mean for the number of days during past 30 days when a person's mental health was not good is contained inside the interval (3.32, 4.55) in 2010 b) It means that if we were to collect say total 100 samples each one of size 1151 from the survey, and build 100 confidence intervals each one of 95% conf level, then about 95 of those intervals will have the true mean value inside them. c) Since Sample size 2755 > current sample size = 1151, it atd error will be smaller because atd err = Vn

(d)

As we decrease the sample size, the standard deviation of proportion increases. Hence,

Standard error of estimate will be larger