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Statistical inference — Question 15

QCAA 2023, Paper 2 · 7 marks

Q15 · 2023 · Technology-activeSimple familiar7 marks

QUESTION 15 (7 marks)

The travel time for students attending a certain university is assumed to be normally distributed, with a population mean of 25.2 minutes and standard deviation of 4.7 minutes. Travel times are collected from a random sample of 120 of these students and used to calculate a sample mean, Xˉ1\bar X_1, in minutes.
a)
Determine P(Xˉ1≤25)P(\bar X_1\le25).
[2 marks]
b)
Given P(Xˉ1>k)=0.9P(\bar X_1>k)=0.9, determine the value of kk.
[1 mark]
c)
Travel times are collected from a second random sample of the university’s students and used to calculate a second sample mean, Xˉ2\bar X_2, in minutes. Given P(Xˉ2≤25)≈0.4P(\bar X_2\le25)\approx0.4, determine the number of students in the second sample.
[4 marks]
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