Question

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1. 1. For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg (based on data from the National Health Survey). Hypertension is commonly defined as a systolic blood pressure above 140 mm Hg.

1. If a woman between the ages of 18 and 24 is randomly selected, find the probability that her systolic blood pressure is greater than 140.
2. If 4 women in that age bracket are randomly selected, find the probability that their mean systolic blood pressure is greater than 140.
3. Given that part b) involves a sample size that is not larger than 30, why can the central limit theorem be used?
4. If a physician is given a report stating that 4 women have a mean systolic blood pressure below 140, can she conclude that none of these women has hypertension?

2. The population mean for commute time is 25.1 minutes and the standard deviation is 10.2 minutes in Aurora, CO.

a) Based on a sample size of 35 individuals, determine the two-sided 90% confidence interval for the distribution of commute times for this population.

b) The sample that we selected above has a mean commute time of 35 minutes. Based on our confidence interval, does the sample of individuals appear to be representative of the population? Why or why not?

2. We want to determine whether UCCS students who are successful in getting into grad school have higher GRE scores compared to all UCCS students who took the GRE exam. We select a random sample (n=20) of successful grad school applicants from UCCS. Based on our sample, the mean is 550 and the standard deviation is 60. The population mean for all UCCS students who took the exam was 480. GRE scores are normally distributed.    

a) What would be the 2-sided 99% confidence interval?

b) Based on the 99% confidence interval, do successful grad school applicants appear to have a higher average score on the GRE exam compared to all UCCS students who took the exam? Why or why not?

3. In an effort to determine whether exposure to high lead levels has an effect on blood pressure in young children, blood-pressure measurements were taken on 30 children aged 5-6 years living in a specific community exposed to high lead levels. For these children, the mean diastolic blood pressure was found to be 66.2 mm Hg with standard deviation 7.9 mm Hg. From a nationwide study, we know that the mean diastolic blood pressure is 58.2 mm Hg for 5- to 6-year old children. We will assume that exposure to lead will have either no effect or cause an increase in blood pressure. Determine a one-sided 95% confidence interval for diastolic blood pressure among 5- to 6-year-old children in this community based on the observed 30 children. Based on this confidence interval, does it appear that children who are exposed to lead have higher blood pressure?

Answer 1

1)

a)

µ =    114.8                                      
σ =    13.1                                      
n=   1                                      
                                          
X =   140                                      
                                          
Z =   (X - µ )/(σ/√n) = (   140   -   114.8   ) / (    13.1   / √   1   ) =   1.924  
                                          
P(X ≥   140   ) = P(Z ≥   1.92   ) =   P ( Z <   -1.924   ) =    0.0272           (answer)

b)

µ =    114.8                                      
σ =    13.1                                      
n=   4                                      
                                          
X =   140                                      
                                          
Z =   (X - µ )/(σ/√n) = (   140   -   114.8   ) / (    13.1   / √   4   ) =   3.847  
                                          
P(X ≥   140   ) = P(Z ≥   3.85   ) =   P ( Z <   -3.847   ) =    0.0001           (answer)

c)

because population from which sample is taken is normally distributed, so, central limit theorem applies on small sample size

d)

NO, mean can be less than 140 but individual values may be above 140