Question

# 1.A weekend TV game show called rolling a dice is running each week. For each roll,...

1.A weekend TV game show called rolling a dice is running each week. For each roll, if the dice shows an odd number the participate earns 5 dollars and otherwise, he gets nothing. Each participate can roll the dice 20 times. Let X denote the number times the dice shows an odd number.
a) (2 mark) Calculate the average earning of a participate.
b) Calculate the probability that ? ⩽ 2 (show your final answer correct to four decimal places).
c) Under appropriate condition, a Binomial random variable X can be approximated by a Normal random variable Y with the same mean and standard deviation. In doing so, we need to apply a continuity correction. That is
? (? ⩽ ?) ≈ ? (? ⩽ ? + 0.5).
Calculating the probability that ? ≤ 6 by using Normal approximation with
continuity correction.
2. A manufacturer is interested in estimating the lifetime of printers. A confidence interval is constructed based on a sample of 20 printers and the assumption that the population distribution of the lifetime of a printer is normal.
a) If the population standard deviation is 1.5 and the upper end of a 95% confidence interval is 6.36, find the observed sample mean (show your final answer correct to two decimal places).
b) If, instead, we only know the sample standard deviation (which is 1.2), and given the lower end of a 90% confidence interval (which is 3.14), find the observed sample mean (show your final answer correct to two decimal places).
c) Given the significance level β= 0.1, and the sample standard deviation is s = 1.2, perform a hypothesis testing (with detailed steps) to test if the population variance ?2 = 1.8.

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