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2022 T1 EXAM
In 2022, a wine seller wants to open a new shop in the local area and needs to borrow money from his bank. Hoping to convince his banker, he polls a random sample of 75 people among which 59 declare they drink red wine. He then meets his banker with this information and also tells him that in September 2012, a national newspaper reported that \( 63 \% \) of the Australian adult population drank red wine. Suppose for this question that this was the true proportion of Australians drinking red wine, as of September 2012.
Let \( X_{1} \) be the number of Australian adults drinking red wine in a random sample of 25 people gathered in 2012. Let \( X_{2} \) be the number of Australian adults drinking red wine in a random sample of 75 people collected in 2022.
(a) Decide upon the exact correct distribution of \( X_{1} \) among the choices presented below:
\( N(2012,63) \)
\( t(24) \)
\( \chi^{2}(24) \)
\( t(25) \)
\( B(25,63) \)
\( B(25,0.63) \)
\( \chi^{2}(1) \)
\( N(0,1) \)
(b) Using the distribution picked in the part (a), what is the probability at least 12 people drank red wine in the sample of 25 people gathered in 2012 ?
\( \square \) (enter your answer correct to at least 3 significant figures.)
(c) Using the normal approximation and the continuity correction, what is the probability at least 12 people drank red wine in the sample of 25 people gathered in 2012 ? Choose the correct answer from the list below.
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