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Jack decides to get a new tight haircut. He constructs a one-sample t-interval to estimate the average number of seconds that pass by before someone comments on his hair. He randomly selects $40$ different lengths of time it takes for someone to comment on his new haircut. After constructing the interval he wants to get a better estimate with the margin of error only one-half as large.

What should the new sample size be?

A

$10$

B

$20$

C

$80$

D

$120$

E

$160$

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