Download systolic blood pressure

SYSTOLIC BLOOD PRESSURE
Math 116 – Reviewing for the Final Exam
This is all about Means
Chapter 6 – Normal Distributions
1) For 18-24 year old women, systolic blood pressures (in mm Hg) are normally distributed with
a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg.
a) Identify the population and the variable.
Population: 18-24 year old women
Variable: systolic blood pressures (in mm Hg)
b) Identify usual and unusual systolic blood pressures for this population according to the
range rule of thumb.
  2  114.8  2 *13.1
[88.6,141]
Any systolic blood pressure reading which is outside of the above interval is considered to
be unusual
c) If one woman from this age group is randomly selected, what is the probability that her
systolic blood pressure is
(i)
Between 94 and 142 mm Hg?
94  114.8
142  114.8
z
)
13.1
13.1
P(1.59  z  2.07)  0.9808  0.0559  .9249
P(94  x  142)  P(
With calculator: normalcdf(94, 142, 114.8, 13.1) = .9249
(ii)
P( x  82)  P( z 
At most 82 mm Hg?
82  114.8
)  P( z  2.50)  0.0062
13.1
With calculator: normalcdf(-10^9, 82, 114.8, 13.1) = 0.0061
(iii)
P( x  143)  P( z 
At least 143 mm Hg?
143  114.8
)  P( z  2.15)  1  .9842  .0158
13.1
With calculator: normalcdf(143, 10^9, 114.8, 13.1) = 0.0157
d) Find the two blood pressures having these properties: the mean is midway between them
and 90% of all blood pressures are between them.
There are two scores, one separating the top 5% (with an area of .95 to its left), and
another separating the bottom 5% (with an area of 0.05 to its left). Use the table, from
inside out to find the z scores. The z-scores are – 1.645, and 1.645. Then, use the formula:
x    z  114.8  1.645*13.1  93.25
x    z  114.8  1.645*13.1  136.35
With calculator: invNorm(0.05, 114.8, 13.1) = 93.25
With calculator: invNorm(0.95, 114.8, 13.1) = 136.35
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SYSTOLIC BLOOD PRESSURE
Chapter 7 – Section 7.2 – Distribution of Sample Means
2) For women aged 18-24, systolic blood pressures (in mm Hg) are normally distributed with a
mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg.
a) Give the shape, mean and standard deviation of the distribution of sample means for samples
of size 36.
X is Normal with   114.8 and   13.1
According to the Central Limit Theorem, the distribution of sample means, for samples of
size 36, is also normally distributed with
 x    114.8
x 

n

13.1
 2.18
36
b) What is the probability that a sample of 36 women of this age group has a mean systolic blood
pressure of at least 121 mm Hg?
121  114.8
)  P( z  2.84)  1  .9977  .0023
13.1
36
13.1
2
With calculator: normalcdf(121, 109 , 114.8,
) = 0.0023~0.002=
1000
36
P( x  121)  P( z 
c) If the population mean is 114.8, the probability of obtaining a sample of size 36 with a mean
of 121 or more is __0.002_____.
So, for samples of size 36, about __2__ samples in 1000 will have a sample mean of 121 or more
when the population mean is 114.8.
Because this event only happens _2__ out of __1000____ times, we consider it to be
usual/unusual.
d) What may this result suggest?
If the population mean is 114.8, it’s very unusual to observe a sample of size 36 with a
mean of 121 or more. This unusually high result may suggest that “probably” the sample
was selected from a population with mean higher than 114.8.
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SYSTOLIC BLOOD PRESSURE
Chapter 8 – Confidence Intervals about a Population Mean μ
3) We want to estimate the mean systolic blood pressure of a group of overweight 18-24 year old
women. Thirty-six women from this group were selected at random and their mean systolic
blood pressure was 121 mm Hg. Assume systolic blood pressures of women of this age group
have a standard deviation of 13.1 mm Hg.
a) What is the point estimate?
The point estimate is the sample mean, that is x-bar = 121
b) Verify that the requirements for constructing a confidence interval about x-bar are
satisfied.
 The sample is a simple random sample
 The value of the population standard deviation σ is known. (we’ll use z)
 The sample size is larger than 30
c) Construct a 99% confidence interval estimate for the systolic blood pressure of all
overweight women of this age group. (Are you using z or t? Why?)
The value of the population standard deviation σ is known. (We’ll use z)
x z*

n
   x z*

n
13.1
13.1
   121  2.575*
36
36
121  5.622    121  5.622
121  2.575*
115.378    126.622
For calculator feature use STAT, arrow to TESTS, and select 7:ZInterval,
select Stats enter the required information, and CALCULATE
d) The statement “99% confident” means that, if 100 samples of size __36___ were taken,
about __99___ intervals will contain the parameter μ and about __1__ will not.
e) We are __99___% confident that the mean systolic blood pressure of overweight 18-24
year old women is between ___115.38__ and ____126.62 mm Hg__
f) With 99% confidence we can say that the mean systolic blood pressure of overweight 1824 year old women is ___121 __ with a margin of error of __5.62_____
g) For 99% of such intervals, the sample mean would not differ from the actual population
mean by more than __5.62 mm Hg_____
h) What would be necessary in order to construct a more precise 99% confidence interval
estimate for the systolic blood pressure of this group?
Select a larger sample from the population, or use a lower confidence level.
3
i) You know that the mean systolic blood pressure of women aged 18-24 is 114.8 mm Hg.
What does the interval constructed in part (c) suggest? Explain.
The interval (115.38, 126.62) is completely above 114.8, which suggests, with 99%
confidence, that the mean systolic blood pressure of the group of women from which the
sample was selected is higher than 114.8 mm Hg.
j) How large of a sample should be selected in order to be 99% confident that the point
estimate x-bar will be within 4 units of the true population mean?
 z *    2.575*13.1 
n
 
  72
4
 E  

2
2
k) Circle the correct choice:
 Increasing the confidence level produces a longer/shorter
 Increasing the confidence level
increases/decreases
 Increasing the sample size
increases/decreases
interval.
the precision.
the precision.
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SYSTOLIC BLOOD PRESSURE
Chapter 9 – Testing a Mean μ
4) For women aged 18-24, systolic blood pressures (in mm Hg) are normally distributed with a
mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. A researcher claims that
overweight women have a higher systolic blood pressure. Thirty-six women from a group of
overweight women from this age group were selected at random and their mean systolic blood
pressure was 121 mm Hg. Test the researcher’s claim at the .5% level of significance. (Are you
using z or t? Why?)

Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the
point estimate in the graph.
****You should be wondering: Is x-bar = 121, higher than 114.8 by
chance, or is it significantly higher? The p-value found below will help
you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and
the results:
Test statistic =
p-value =
***How likely is it observing an x-bar = 121 or more when you select a
sample of size 36 from a population that has a mean µ of 114.8?
very likely,
likely,
unlikely,
very unlikely
*** Is x-bar higher than 114.8 by chance or significantly higher?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
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SYSTOLIC BLOOD PRESSURE
Chapters 8 and 9 – Hypothesis Testing and Confidence Intervals for 1  2
(Independent Samples)
5) A researcher wishes to determine whether people with high blood pressure can reduce their
blood pressure by following a particular diet. Use the sample data below to test the claim that the
treatment population mean 1 is smaller than the control population mean  2 . Test the claim
using a significance level of 0.01.
Treatment
Control
Sample size
Mean
85
75
189.1
203.7
Sample Standard
deviation
38.7
39.2
a) Test the claim at the 1% level of significance. (Are you using z or t? Why?)
 Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the
point estimate in the graph.
****You should be wondering: Is the difference between the x-bars
lower than zero by chance, or is it significantly lower? The p-value
found below will help you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and
the results:
Test statistic =
p-value =
***How likely is it observing such a difference between the x-bars (or a
more extreme one) when the mean of the two populations is equal?
very likely,
likely,
unlikely,
very unlikely
*** Is the difference between the x-bars lower than zero by chance, or is
it significantly lower?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
b) Construct a 98% confidence interval estimate for the mean difference between the blood
pressures of the two groups. What does the interval suggest? (Are you using z or t?
Why?)
6
SYSTOLIC BLOOD PRESSURE
Chapter 9 – Testing 1  2 (Dependent Samples)
6) A researcher wishes to determine whether people with high blood pressure can reduce their
blood pressure by following a particular diet. 85 people with high blood pressure were selected at
random and the before the diet and after the diet blood pressures were compared. The mean of
the differences was –11.9 and the standard deviation 54.1. Test the claim using a significance
level of 0.01.
a) At the 0.01 significance level can we conclude that the diet helped reduce the blood pressure?
 Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the
point estimate in the graph.
****You should be wondering: Is the sample mean difference d-bar = 11.9 lower than zero by chance, or is it significantly lower? The p-value
found below will help you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and
the results:
Test statistic =
p-value =
***How likely is it observing such a value of d-bar (or a more extreme
one) when the population mean difference is zero?
very likely,
likely,
unlikely,
very unlikely
*** Is the mean difference d-bar = -11.9 lower than zero by chance, or is
it significantly lower?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
b) Construct a 98% confidence interval of the population mean difference. What does the
interval suggest?
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