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A consumer group selected 100 different airplanes at random from each of two large airlines. The mean seat width for the 100 airplanes was calculated for each airline, and the difference in the sample mean widths was calculated. The group used the sample results to construct a 95 percent confidence interval for the difference in population mean widths of seats between the two airlines. Suppose the consumer group used a sample size of 50 instead of 100 for each airline. When all other things remain the same, what effect would the decrease in sample size have on the interval

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Answer:

Decreasing the sample size decreases the width of the confidence interval.

Step-by-step explanation:

-Let s be the sample standard deviation and [tex]\bar X[/tex] be the sample mean.

#We calculate the 95% confidence interval for the sample size 100 as below:

[tex]\bar X\pm z_{0.05/2}\frac{s}{\sqrt{n}}\\\\\\=\bar X\pm 1.960\times\frac{s}{\sqrt{50}}\\\\\\=\bar X\pm 0.1960s[/tex]

#we then calculate the confidence interval of sample size =50 and of similar mean and standard deviation:

[tex]\bar X\pm z\frac{s}{\sqrt{n}}\\\\\\=\bar X\pm z_{0.05/2}\times\frac{s}{\sqrt{n}}\\\\\\=\bar X\pm1.960\times \frac{s}{\sqrt{50}}\\\\\\\\=\bar X\pm0.2772s[/tex]

#Notice that the smaller sample(n=50) has a narrower confidence interval.

Hence, decreasing the sample size decreases the width of the confidence interval.

Answer: B The width of the interval would increase

Step-by-step explanation:

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