Question

In: Statistics and Probability

Many high school students take the AP tests in different subject areas. In 2007, of the...

Many high school students take the AP tests in different subject areas. In 2007, of the 143044 students who took the AP biology exam 76712 of them were female. In that same year, of the 211993 students who took the AP calculus AB exam 100106 of them were female. Is there enough evidence to show that the proportion of AP biology exam takers who are female is higher than the proportion of AP calculus AB exam takers who are female?

a) Test at the 5% level

b) Compute a 90% confidence interval for the difference in proportions.

Use the steps of PHANTOMS for the hypothesis test.

For the confidence interval you do not need to do all the steps of PANIC since you did some of them already in PHANTOMS.

You just need to do the NIC of PANIC.

Part a.) HYPOTHESIS TEST

P: Parameter

What is the correct parameter symbol and wording for population 1?

     Select an answer p̂₁ N₁ n₁ p₁ μ? X̄ μ₁  = Select an answer A randomly selected student who took the AP biology test that is female The percentage of all students who took the AP biology test that are female A randomly selected student who took the AP biology test 143044 randomly selected students who took the AP biology test All students who took the AP biology test The percentage of 143044 randomly selected students who took the AP biology test that are female All students who took the AP biology test that are female Whether or not a randomly selected student who took the AP biology test is female

     What is the correct parameter symbol and wording for population 2?

     Select an answer μ? p̂₂ N₂ μ₂ p₂ X̄₂ n₂  = Select an answer All students who took the AP calculus AB test that are female The percentage of all students who took the AP calculus AB test that are female All students who took the AP calculus AB test A randomly selected student who took the AP calculus AB test The percentage of 211993 randomly selected students who took the AP calculus AB test that are female A randomly selected student who took the AP calculus AB test that is female Whether or not a randomly selected student who took the AP calculus AB test is female 211993 randomly selected students who took the AP calculus AB test

H: Hypotheses

Fill in the correct null and alternative hypotheses:


H0:H0: Select an answer μ₁ - μ₂ p₁ - p₂ X̄₁ - X̄₂ N₁ - N₂ p̂₁ - p̂₂ n₁ - n₂ μ?  ? > ≥ = ≠ ≤ <  

HA:HA: Select an answer μ₁ - μ₂ n₁ - n₂ X̄₁ - X̄₂ p̂₁ - p̂₂ N₁ - N₂ p₁ - p₂ μ?  ? ≠ ≥ = > < ≤  


A: Assumptions

Since Select an answer quantitative qualitative  information was collected from each object, we need to check the following conditions:

Check all that apply.

    

  • Normal population or at least 30 pairs of data with no outliers in the differences
  • σσ is unknown for each group.
  • The samples are dependent.
  • n1−x1≥10n1-x1≥10 and n2−x2≥10n2-x2≥10
  • The samples are independent.
  • x1≥10x1≥10 and x2≥10x2≥10
  • Normal population or n1≥30n1≥30 and n2≥30n2≥30 with no outliers for each group.
  • N1≥20n1N1≥20n1 and N2≥20n2N2≥20n2



     Check those assumptions:

x1x1 =  which is ? ≥ ≠ = < > ≤  

x2x2 =  which is ? > ≥ < ≤ = ≠  

n1−x1n1-x1 =  which is ? ≥ = < ≠ > ≤  

n2−x2n2-x2 =  which is ? = ≤ > < ≠ ≥  

Population sizes are not given.

We will assume that N1N1 >= 20(n1)n1) and N2N2 >= 20(n2)n2)


N: Name the test

The conditions are met to use a Select an answer T-Test Paired T-Test 2-Proportion Z-Test 2-Sample T-Test 1-Proportion Z-Test  .

T: Test Statistic

The symbol and value of the random variable (to 4 decimal places) on this problem are as follows:


     Select an answer n₁ - n₂ N₁ - N₂ μ₁ - μ₂ p₁ - p₂ μ? p̂₁ - p̂₂ X̄₁ - X̄₂  =

Pooled Sample proportion of ˆpp^ is as follows:

     (Leave your answer in FRACTION form and use this fraction form in the set up of the test statistic)

ˆpp^ = x1+x2n1+n2x1+x2n1+n2 =

(( ++  )) /(/(  ++  )=)=

Set up the formula for the test statistic with EXACT FRACTIONS or given decimal values for each box:
z=ˆp1−ˆp2√ˆp(1−ˆp)(1n1+1n2)=z=p^1-p^2p^(1-p^)(1n1+1n2)=

((  −-  ) / √(( ⋅⋅ (1−(1- )⋅ (1)⋅ (1/  +1+1/  ))=))=

Round final answer from technology to 2 decimal places.

     z =

O: Obtain the P-value

Report the final answer to 4 decimal places. It is possible when rounded that a p-value is 0.0000

     P-value =

M: Make a decision

Since the p-value ? = ≤ ≠ ≥ < >   , we Select an answer accept H₀ reject Hₐ fail to reject H₀ fail to reject Hₐ reject H₀  .

S: State a conclusion

  • There Select an answer is not is  significant evidence to conclude Select an answer All students who took the AP biology test that are female Whether or not a randomly selected student who took the AP biology test is female 143044 randomly selected students who took the AP biology test The percentage of all students who took the AP biology test that are female A randomly selected student who took the AP biology test The percentage of 143044 randomly selected students who took the AP biology test that are female All students who took the AP biology test A randomly selected student who took the AP biology test that is female  Select an answer is more than is less than is equal to differs from  Select an answer 211993 randomly selected students who took the AP calculus AB test The percentage of all students who took the AP calculus AB test that are female The percentage of 211993 randomly selected students who took the AP calculus AB test that are female All students who took the AP calculus AB test that are female A randomly selected student who took the AP calculus AB test Whether or not a randomly selected student who took the AP calculus AB test is female All students who took the AP calculus AB test A randomly selected student who took the AP calculus AB test that is female

Part b.) CONFIDENCE INTERVAL

N: Name the procedure

   The conditions are met to use a Select an answer Paired T-Interval 2-Sample T-Interval T-Interval 2-Proportion Z-Interval 1-Proportion Z-Interval

I: Interval estimate (round endpoints to 3 decimal places)

A  % confidence interval for Select an answer p₁ - p₂ μ? μ₁ - μ₂ n₁ - n₂ X̄₁ - X̄₂ p̂₁ - p̂₂ N₁ - N₂  is (  ,   )

C: Conclusion in context

  • We are  % confident that Select an answer All students who took the AP biology test that are female Whether or not a randomly selected student who took the AP biology test is female 143044 randomly selected students who took the AP biology test All students who took the AP biology test The percentage of 143044 randomly selected students who took the AP biology test that are female A randomly selected student who took the AP biology test A randomly selected student who took the AP biology test that is female The percentage of all students who took the AP biology test that are female  is between  % and  % Select an answer more than less than  Select an answer A randomly selected student who took the AP calculus AB test that is female Whether or not a randomly selected student who took the AP calculus AB test is female All students who took the AP calculus AB test that are female All students who took the AP calculus AB test The percentage of all students who took the AP calculus AB test that are female A randomly selected student who took the AP calculus AB test 211993 randomly selected students who took the AP calculus AB test The percentage of 211993 randomly selected students who took the AP calculus AB test that are female

LicensePoints possible: 58
This is attempt 1 of 2.

Solutions

Expert Solution

a)

paramter

p1 = proportion of AP biology exam takers who are female

p2 = proportion of AP calculus AB exam takers who are female

X1 =76712, n1 = 143044

X2 = 100106 , n2 = 211993

p^ = (X1+x2)/(n1+n2) = 0.49802697

Ho:p1 = p2

Ha: p1 > p2

TS = 37.45

p-value = P(Z >TS) = 0.000

since p-value < alpha

we reject the null hypothesis

we conclude that there is evidence that

proportion of AP biology exam takers who are female is higher than the proportion of AP calculus AB exam takers who are female

b)

90% confidence interval

90% CI is (0.0613,0.0669


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