New drug Sample size
Number of people with heart attack using the new
Proportion of heart attacks with new drug 2219
26
0.0117 Placebo Sample size
Number of people with heart attack using Placebo
Proportion of heart attacks with Placebo 2035
46
0.0226 Difference between both drugs 0.0109 Hypothesized Value 0 Standard Error
Test Statistic z 0.0040
2.72 p-value (TwoTail) 0.66% H0 : Proportion of Placebo - Proportion of ne
Ha : Proportion of Placebo - Proportion of ne Proportion of heart attacks for the placebo was 0.0226. T
group was 0.0117. Therefore, difference is 0.0109. The hypothesized value is 0, because this is a two tailed t
=C7-C3
Given That leaves us with p-hat for the placebo and new drug,
When we calculate the standard error, we get 0.004.
Therefore, our Z-statistic is 2.725. =SQRT((C3*(1-C3)/C1)+(C7*(1-C7)/C5))
=C9/C13
=2*(1-NORMSDIST(C14)) So let's see what this looks like. We have our normal dist
difference between the two populations would be zero. I
had a lower proportion of people that suffered heart atta
0.0109. This is a two-tailed test with a 95% significance level, wit The Z-score for this confidence level would be 1.96. This
1.96 standard deviations from the expected outcome, th Our result was 2.725 standard deviations from the expec
chart, 2.725 corresponds with an outcome that is 0.66% We have to reject our null hypothesis. In other words, we
exhibited by the group that took the new drug did not oc Note: Hypothesis rule is:
If your P-Value is ≤ 0.05, you will reject the null hypothei o - Proportion of new drug = 0
o - Proportion of new drug ≠ 0 he placebo was 0.0226. The proportion of heart attacks for the new drug
fference is 0.0109. ause this is a two tailed test.
placebo and new drug, which again, are 0.0117 and 0.0226 respectively.
error, we get 0.004. 5. We have our normal distribution. If the null hypothesis were true, the
ulations would be zero. In our samples, we found that the new drug group
e that suffered heart attacks. The difference between the two groups was 5% significance level, with an alpha of 0.05 or 5%. evel would be 1.96. This means that if our actual outcome were more than
e expected outcome, then we must reject our null hypothesis. eviations from the expected outcome. In fact, by looking at a Z distribution
n outcome that is 0.66% likely. hesis. In other words, we can feel fairly confident that the positive results
the new drug did not occur by chance.
reject the null hypotheis London Heathrow
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39 B C D E Hypothesis Test About a Population Mean
With σ Unknown
Sample Size
Sample Mean
Sample Std. Deviation
Hypothesized Value 7 Standard Error
Test Statistic t
Degrees of Freedom
p-value (Upper Tail) Businessweek did a survey of airports according to the mean rating given by business tra
Each airport was rated on a scale of 0 to 10 with 10 being the highest score. The rating fo
London's Heathrow Airport are given to the left..
A score of 7 or more will be considered superior.
Develop the hypothesis that London's Heathrow is superior. Use the equations above to determine sample size, sample mean, sample SD, etc. Should we reject or fail to reject the null hypothesis? What conclusion can we draw from our data? Page 4 London Heathrow
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8 Page 5 E London Heathrow
F G H I J 1 Your hypothesis would be: 2
3
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5
6 =COUNT(A2:A61)
=AVERAGE(A2:A61)
=STDEV.S(A2:A61) 7
8 Given 9 H0 : µ of Heathrow ≤ 7
Ha : µ of Heathrow > 7
Follow the instructions to find your P-Value
If your P-Value is ≤ 0.05, you will reject the null hypotheis =D6/SQRT(D4)
11 =(D5-D8)/D10
12 =D4-1
10 13
14 =T.DIST.RT(D11,D12) 15
16
17 e mean rating 18
given by business travelers.
19score. The rating for
ing the highest
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mple mean, sample
SD, etc.
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61 Page 7 I J London Heathrow
K L M N 1
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6 our P-Value 7
8 hypotheis
reject the null
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K L M N 40
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61 Page 9 O Mew = population mean = μ
Sigma = population standard deviation = s
Standard Error = Standard Deviation for the Sampling distribution of Xbar = SE
Hypothesized Value of Population Mean = μ0 = Assumed Value of Population Mean used for testing pro
sample size = n
Sample Mean = Xbar
Sample Standard Deviation (for when sigma not known) = s
Alpha = a = Risk of rejecting (Original Statement) H 0 when it is actually true = Type 1 Error Test Statistic = Used to determine whether to reject H 0 and accept Ha = Number of SE above or below hy Use z test statistic if Sigma known or can be estimated, or you are significance Testing a Proportion a
Use t test statistic if Sigma NOT known or can be estimated A statement from an official report said that Realtors make $85,000 a year (national mean). Researcher thin
Realtors in the Des Moines area have a mean annual salary of more than 85,000 a year. At alpha = .05, sig
$12,549, n = 36 and sample mean = $88,595, can we conclude that realtors in the Des Moines area make m
than $85,000? Sample Salaries
$98,356.00
$120,492.00
$119,398.00
$92,682.00
$113,284.00
$79,398.00
$82,503.00
$81,485.00
$84,083.00
$76,384.00
$86,539.00
$69,894.00
$67,810.00
$84,219.00
$90,242.00
$104,294.00
$86,642.00
$85,646.00
$73,286.00
$65,074.00 Hypothesis Test (Significance Test)
Point of view:
Researcher who believes that Des Moines realtors have an annual mean
Considering:
Population of Des Moines Realtors
Goal:
Run Hypothesis test to provide statistical evidence to support the claim
Realtors have an annual mean salary of more than $85,000. Mew > 8500
Step 1: List H0 and Ha Step 2: Select Alpha Alpha = Alpha = a = Risk of rejecting H0 when it is actually true At alpha = 0.05, 5% risk of rejecting H 0 even though it Step 3: Draw Picture, Collect Data, Calculate Sample Statistics, Calculate Test Statistic
Hypothesized Mean = μ0 =
Sigma = s =
Sample SD = s
Test Statistic To Use:
sample size = n
Sample Mean = Xbar
=AVERAGE(values)
Alpha = a =
Type of Test
SE
Standard Error = Standard Dev
=(Xbar - μ0)/SE
Test Statistic = =
Sample Error in numerator, Sta
Step 4: Create p-value Rejection Rule and calculate p-value
p-value <= alpha, Reject H 0 and accept Ha , otherwise Fail to Reject H0
Rejection Rule:
p-value One Tail To Right =1-NORM.S.DIST(z,1) $77,829.00
$106,881.00
$83,221.00
$88,483.00
$80,998.00
$84,734.00
$126,949.00
$64,085.00
$81,469.00
$85,769.00
$98,864.00
$92,132.00
$65,025.00
$101,594.00
$91,250.00
$98,426.00 Step 4: Calculate Critical Value and Critical Value Rejection Rule
Test Statistic >= Critical Value, Reject H0 and accept Ha , otherwise Fail to Reje
Rejection Rule:
Critical Value One Tail To Right =NORM.S.INV(1-alpha) Step 5: Write Conclusion Because the p-value of 0.043 <= alpha of 0.05, we reject H0 and accept Ha.
Because the test statistic of 1.72 >= critical value of 1.645, we reject H0 and accept Ha.
The statistical evidence suggests that the mean annual salary for realtors in Des Moines is
greater than the national mean.
At an alpha of 0.05, our sample mean of $88,595 provides statistically significant evidence that
the mean annual salary for realtors in Des Moines is greater than the national mean.
We do run a 5% risk of a Type 1 Error that we might say that the mean annual salary for
realtors in Des Moines is greater than $85,00, when it is not greater than $85,000. One Tail To Lef
One Tail To Right
Two Tail n Mean used for testing procedure er of SE above or below hypothesized mean ance Testing a Proportion and all 4 binomial tests have been met onal mean). Researcher thinks that
000 a year. At alpha = .05, sigma =
n the Des Moines area make more altors have an annual mean greater than $85,000 dence to support the claim that Des Moines
e than $85,000. Mew > 85000 >>> 1 tail right ng H0 when it is actually true = Type 1 Error f rejecting H 0 even though it was true, if we reject Mew <= $85,000, about 5 out of 100 times we will be incorrect. andard Error = Standard Deviation for the Sampling distribution of Xbar = SE ample Error in numerator, Standard Deviations in Denominator = "How Many SD above or below Hypothesized Pop Mean wise Fail to Reject H0
Probability of getting Test Statistic or higher ept Ha , otherwise Fail to Reject H0
This is the hurdle Point and accept Ha.
n Des Moines is nificant evidence that nual salary for e incorrect. Mew = population mean = μ
Sigma = population standard deviation = s
Standard Error = Standard Deviation for the Sampling distribution of Xbar = SE
Hypothesized Value of Population Mean = μ0 = Assumed Value of Population Mean used for testing pro
sample size = n
Sample Mean = Xbar
Sample Standard Deviation (for when sigma not known) = s
Alpha = a = Risk of rejecting (Original Statement) H 0 when it is actually true = Type 1 Error Test Statistic = Used to determine whether to reject H 0 and accept Ha = Number of SE above or below hy Use z test statistic if Sigma known or can be estimated, or you are significance Testing a Proportion a
Use t test statistic if Sigma NOT known or can be estimated A statement from an official report said that Realtors make $85,000 a year (national mean). Researcher thin
Realtors in the Des Moines area have a mean annual salary of more than 85,000 a year. At alpha = .05, sig
$12,549, n = 36 and sample mean = $88,595, can we conclude that realtors in the Des Moines area make m
than $85,000? .
.
.
.
Sample Salaries Hypothesis Test (Significance Test)
Point of view:
Researcher who believes that Des Moines realtors have an annual mean grea
Considering:
Population of Des Moines Realtors
Goal:
Run Hypothesis test to provide statistical evidence to support the claim that D
Realtors have an annual mean salary of more than $85,000. Mew > 85000 >>
Step 1: List H0 and Ha $98,356.00
$120,492.00
$119,398.00
$92,682.00
$113,284.00
$79,398.00
$82,503.00
$81,485.00
$84,083.00
$76,384.00
$86,539.00
$69,894.00
$67,810.00
$84,219.00
$90,242.00
$104,294.00
$86,642.00
$85,646.00
$73,286.00 Step 2: Select Alpha H0 : μ <= $85,000 Ha : μ > $85,000 Alpha = 0.05 Alpha = a = Risk of rejecting H0 when it is actually true
At alpha = 0.05, 5% risk of rejecting H 0 even though it Step 3: Draw Picture, Collect Data, Calculate Sample Statistics, Calculate Test Statistic
Hypothesized Mean = μ0 =
$85,000.00
Sigma = s =
$12,549.00
Sample SD = s
NA
Test Statistic To Use:
z
Because Sigma Known
sample size = n
36
Sample Mean = Xbar
$88,595.00
=AVERAGE(values)
Alpha = a =
0.05
Type of Test
One Tail To Right
SE
2091.5
=Sigma/SQRT(n)
Standard Error = Standard Dev
=(Xbar - μ0)/SE
Test Statistic = z =
1.7188620607
Sample Error in numerator, Sta
Step 4: Create p-value Rejection Rule and calculate p-value
p-value <= alpha, Reject H 0 and accept Ha , otherwise Fail to Reject H0
Rejection Rule:
p-value One Tail To Right 0.0428197459 =1-NORM.S.DIST(z,1) $65,074.00
$77,829.00
$106,881.00
$83,221.00
$88,483.00
$80,998.00
$84,734.00
$126,949.00
$64,085.00
$81,469.00
$85,769.00
$98,864.00
$92,132.00
$65,025.00
$101,594.00
$91,250.00
$98,426.00 Step 4: Calculate Critical Value and Critical Value Rejection Rule
Test Statistic >= Critical Value, Reject H0 and accept Ha , otherwise Fail to Reje
Rejection Rule:
Critical Value One Tail To Right .
.
.
.
.
.
.
. 1.644853627 =NORM.S.INV(1-alpha) Step 5: Write Conclusion
Because the p-value of 0.043 <= alpha of 0.05, we reject H0 and accept Ha.
Because the test statistic of 1.72 >= critical value of 1.645, we reject H0 and accept Ha.
The statistical evidence suggests that the mean annual salary for realtors in Des Moines is
greater than the national mean.
At an alpha of 0.05, our sample mean of $88,595 provides statistically significant evidence that
the mean annual salary for realtors in Des Moines is greater than the national mean.
We do run a 5% risk of a Type 1 Error that we might say that the mean annual salary for
realtors in Des Moines is greater than $85,00, when it is not greater than $85,000. One Tail To Lef
One Tail To Right
Two Tail n Mean used for testing procedure er of SE above or below hypothesized mean ance Testing a Proportion and all 4 binomial tests have been met onal mean). Researcher thinks that
000 a year. At alpha = .05, sigma =
n the Des Moines area make more rs have an annual mean greater than $85,000 ce to support the claim that Des Moines
an $85,000. Mew > 85000 >>> 1 tail right ng H0 when it is actually true = Type 1 Error f rejecting H 0 even though it was true, if we reject Mew <= $85,000, about 5 out of 100 times we will be incorrect. andard Error = Standard Deviation for the Sampling distribution of Xbar = SE ample Error in numerator, Standard Deviations in Denominator = "How Many SD above or below Hypothesized Pop Mean wise Fail to Reject H0
Probability of getting Test Statistic or higher ept Ha , otherwise Fail to Reject H0
This is the hurdle Point nt evidence that e incorrect.

 

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