Question

In: Statistics and Probability

For this problem, carry at least four digits after the decimal in your calculations. Answers may...

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of 5380 permanent dwellings on an entire reservation showed that 1564 were traditional hogans.

(a) Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answer to three decimal places.)

lower limit    
upper limit    


Give a brief interpretation of the confidence interval.

1% of the confidence intervals created using this method would include the true proportion of traditional hogans.

99% of all confidence intervals would include the true proportion of traditional hogans.   

99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

1% of all confidence intervals would include the true proportion of traditional hogans.


(c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that is approximately binomial.

Yes, the conditions are satisfied. This is important because it allows us to say that is approximately normal.    

No, the conditions are not satisfied. This is important because it allows us to say that is approximately binomial.

No, the conditions are not satisfied. This is important because it allows us to say that is approximately normal.

Solutions

Expert Solution

Solution :

Given that,

n = 5380

x = 1564

a) Point estimate = sample proportion = = x / n = 1564 / 5380 = 0.2907

1 - = 1 - 0.2907 = 0.7093

b) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z0.005 = 2.576

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 2.576 (((0.2907 * 0.7093) / 5380)

= 0.0159

A 99% confidence interval for population proportion p is ,

± E   

= 0.2907 ± 0.0159

= ( 0.275, 0.307 )

lower limit = 0.275

upper limit = 0.307

99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

c) Yes, the conditions are satisfied. This is important because it allows us to say that is approximately normal.  


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