A researcher is interested in whether birth order has an effect on the size of a

Question

A researcher is interested in whether birth order has an effect on the size of a child’s vocabulary. A vocabulary test is given to 100 children, all age five and all the oldest in their families. Later, the same vocabulary test is administered to the 100 next oldest sibling of these children when they reach five years of age (so both children in each family take the same test at age five). In this study, the oldest child scored higher on the vocabulary test for 63 of the 100 pairs of children (while the younger child scored higher for 37 of the pairs). A researcher wants to determine if this is strong evidence that order of birth has an effect on vocabulary size, or if these data can be explained by chance.

What are the null and alternative hypotheses in this situation?

What is the standard deviation of the proportion of times the older child scores higher if the null hypothesis is true? Explain how you got your answer.

Is the normal approximation appropriate to use in this case? Explain briefly.

What is the test statistic and the calculated p-value? Explain how you got your answer.

What is the conclusion you should draw from these results?

Explanation / Answer

a)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   =   0.5
Ha:   p   =/=   0.5 [ANSWER]

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b)

As we see, the hypothesized po =   0.5      

Getting the standard error of p^, sp, as n = 100,          
          
sp = sqrt[po (1 - po)/n] =    0.05   [ANSWER]  

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c)

Yes, n po and n(1-po) are both greater than 10.

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d)

Getting the point estimate of p, p^,          
          
p^ = x / n =    0.63      
          
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    2.6   [ANSWER, TEST STATISTIC]  
          
As this is a    2   tailed test, then, getting the p value, by table/technology,  
          
p =    0.009322376   [ANSWER]

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e)

As our P value is very small, then this is unlikely due to chance.

There is significant evidence that order of birth has an effect on vocabulary size. [CONCLUSION]

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