1) 
What is meant by a 5% margin of error?
The
difference between the estimate based on the sample percentage and the
actual population percentage is likely to be less than 5% 
2) 
Bob claims that his apples are bigger than Martha's apples.
The mean of Martha's apples is 27. If you perform a significance test, what would the null
and alternative hypothesis be?
H_{o}:
μ = 27, H_{a} μ >27
Note Ho is the null
hypothesis representing "no difference". Ha is the alternative hypothesis
representing "there is a difference". 
3) 
A consultant takes a random sample and gets a
mean size of 28 for Bob's apples based on a sample size of 16. If the
standard deviation of Martha's apples is 4, find the pvalue and answer the
question are Bob's apples larger.
Find the pvalue on the
TI83 as follows:
Note that in this
case we use
s =
σ
/ (16)^{0.5}
= 4 / 4
= 1
pvalue =

2nd 
DISTR normalcdf
( 28, 10000, 27, 1) 





pvalue is always
a tail area and is defined as the probability of getting a test
statistic (in this case 28) as far from the mean or further from the
mean than the one obtained. 

Conclusion: There is not statistically significant (p = 15.9%)
evidence to indicate that Bob's apples are larger than Marta's. 


4) 
What is the best way to establish a cause and effect relationship between two
variables? A controlled experiment.

5) 
Given the equation: yhat = 4x +10, if x is increased by 5 what is the change
in y? 4 x 5 = 20 
6) 
1,000,000 people belong to Martha's Fitness Emporium. 600,000 are women and
400,000 men. 50% of the women work out every week and 40% of the men. In a
sample of 1000, what number would be expected to work out (wo) every week?

7) 
If H_{o}: μ = 50, H_{a} μ >50
and the mean of a sample were 52 with a z score = 2 and a pvalue = 0.023 what
would be the likely conclusion? There is significant reason to conclude that H_{a} μ >50.
Explanation: there is only a 2.3% chance of getting a sample mean as extreme
or more extreme than 52. Therefore it's reasonable to reject the null
hypothesis and accept the alternative hypothesis. The null hypothesis is usually
rejected if p< 5%. 
8) 
Since she wants a proportion
answering a yes or not yes question (not yes includes undecided), we model
the problem binomial. Since she will be surveying a large sample size we can
use a normal approximation for the binomial. All this is a complicated way
of saying that we should use the following equation:
Confidence interval = p ± Z^{* }[ p (1 
p ) / n ]^0.5
Z^{*} = 1.96
M =
Z^{*
}[ p (1  p )
/ n ]^0.5
Unfortunately, we have one
equation and two unknowns (shown in red, knowns are shown in blue), so we're
forced to estimate p. To do so, we want to select the most conservative
possible value and this would be the one giving the highest value of n. It
turns out that p = 50%
always gives the highest value of n
and will always be the one selected when p is unknown.
M^{2 }
= (Z^{*})^{2
} [p (1  p ) / n ]
M^{2} / (Z^{*})^{2
}= p (1  p ) / n 

n 
= 
p (1  p )

M^{2}
/ (Z^{*})^{2} 


n 
= 
0.5 (1  0.5 )

0.02^{2}
/ (1.96)^{2} 


^{ }n
= 2401 Note: always round fractions
upward 



9) 
Bob's TocoRama has a deal. Every
Monday customers can spin a wheel that gives them a discount of up to
$1.00 on their next purchase. The mean is $0.50 and standard deviation is
$0.15. If a customer participates for 40 weeks, what is the probability that
he will have a savings greater than $21?
In 40 weeks the mean savings
will be:
μ = 40 * $0.50
= $20
 Now we can draw a picture of the
problem in order to visualize
 it, but before calculating the area
we have to first determine the standard deviation of the distribution
we've just drawn.
 The standard deviation
for 40 weeks is as follows:
σ
= [40 * (0.15)^{2}]^{0.5}
^{ }
= 0.9487
Note: standard deviations
are not additive, however, variances (the standard deviation squared) are.
Hence, the standard deviations must be squared before being multiplied by
40. This gives the total variance for 40 weeks. Taking the square root of
this this variance give the 40 week standard deviation.
probability = 
2nd 
DISTR normalcdf
( 21, 10000, 20, .9487) 



probability savings greater than $21 = 14.6% 


10) 
10% of Juan's fruit special
cakes weigh less than 20 grams and 15% weigh more than 25 grams. Find the
mean and standard deviation. 



First, draw a picture as shown at left. From this we can find z1 = 
1.28 and z2 = 1.04 using a ztable. Next it's time to write an
equation: 
^{
}z = ( x_{i}^{ } μ
) / σ
Note
that since the above equation has 2 unknowns in it ( μ and σ), we
have to write a version of it for each side as shown below. This
gives 2 equations and 2 unknowns which we can solve as simultaneous
equations.

Left Side 

Right Side 

1.28 = ( 20  μ ) / σ
σ = (  20 + μ ) /
1.28 

1.04
= ( 25  μ ) / σ
σ = ( 25  μ
) / 1.04 
next merge the two
equations as follows:
( μ  20 ) /
1.28
= ( 25  μ ) / 1.04
( μ  20 )
1.04 =
( 25  μ ) 1.28
1.04 μ  20.8
= 32  1.28 μ
1.04 μ + 1.28
μ = 20.8 + 32
2.32 μ = 52.8
μ = 52.8 /
2.32
Substituting μ
into the left side's equation gives
σ = (  20 +
22.76 ) / 1.28
= 2.95 / 0.25





