WACE Maths Methods ATAR Section 1 Topic Tests
Normal Distribution Topic Test 1
Section One: Technology-free
Number of marks: 15
Reading time: 1 minute
Writing time: 15 minutes
Section One:
Answer all questions. Write your answers in the spaces provided.
Supplementary pages for planning/continuing your answers to questions are provided at the end of this Question/Answer booklet. If you use these pages to continue an answer, indicate at the original answer where the answer is continued, i.e. give the page number.
For a set of data values that are normally distributed, approximately 68% of the values will lie within one standard deviation of the mean, approximately 95% of the values will lie within two standard deviations of the mean and approximately 99.7% of the values will lie within three standard deviations of the mean.
If the heights of a large group of women are normally distributed with a mean \(\mu = 163\) cm and standard deviation \(\sigma = 7\) cm, use the above information to answer the following questions:
(a) A statistician says that almost all of the women have heights in the range 142 cm to 184 cm. Comment on her statement. (2 marks)
(b) Approximately what percentage of women in the group has a height greater than 170 cm? (2 marks)
(c) Approximately 2.5% of the women are shorter than what height? (2 marks)
(a) The graphs of three normal distributions are displayed below. The distributions have been labelled A, B and C.
(i) What is the mean of distribution A? (1 mark)
(ii) Which of the distributions has the largest standard deviation? Justify your answer. (1 mark)
(b) A random variable \(X\) is normally distributed. The distribution of \(X\) is graphed below.
(i) Shade the region with area corresponding to \(P(6 \le X \le 9)\). (1 mark)
(ii) Is \(P(6 \le X \le 9) \ge 0.5\)? Justify your answer. (2 marks)
(c) A random variable \(Y\) has probability \(P(Y \ge 2) > P(Y > 2)\). Explain whether it is possible for the distribution of \(Y\) to be normal or binomial. (2 marks)
End of questions
