All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.
(i)
Let
![Rendered by QuickLaTeX.com X](https://theculture.sg/wp-content/ql-cache/quicklatex.com-2453362c766504f3c3806fed710a5337_l3.png)
be the number of people who have blood group A.
![Rendered by QuickLaTeX.com X \sim \text{B}(6, 0.4)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-b20b3139c3af3b4e073e91fac8bdd766_l3.png)
![Rendered by QuickLaTeX.com \text{P}(X = 2) = 0.31104](https://theculture.sg/wp-content/ql-cache/quicklatex.com-281ed8df7b4f25580be5cfa0f63a6482_l3.png)
(ii)
![Rendered by QuickLaTeX.com \text{P}(X \le 2) = 0.54432](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a7676470e0fc38004ae0f5a3d1a43327_l3.png)
Note to leave your answers exactly here.
(i)
![Rendered by QuickLaTeX.com ^8 C_3 \times ^4 C_1 \times ^6 C_2 = 3360](https://theculture.sg/wp-content/ql-cache/quicklatex.com-22c08d71656b1e7e308dd81331fba7d0_l3.png)
(ii)
![Rendered by QuickLaTeX.com ^3 C_1 \times ^5 C_2 + ^5 C_3 = 40](https://theculture.sg/wp-content/ql-cache/quicklatex.com-47676df415948755276f89c4da4935ac_l3.png)
(iii)
Required probability ![Rendered by QuickLaTeX.com = \frac{^5 C_3 \times ^4 C_1 \times ^6 C_2}{3360} = \frac{5}{28}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-83e34ae9c5aab59b20b9ae13fb8e7dfb_l3.png)
(i)
![Rendered by QuickLaTeX.com \text{P}(A \vert B) = \frac{\text{P}(A \cap B)}{\text{P}(B)}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-3bbe334072c3c7db45fed0807d3c43fd_l3.png)
![Rendered by QuickLaTeX.com 0.3 = \frac{\text{P}(A) + \text{P}(B) - \text{P}(A \cup B)}{\text{P}(B)}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-901f01d63451bd8619165fd12c15c314_l3.png)
![Rendered by QuickLaTeX.com 0.3 = \frac{p + 2p - 0.8}{2p}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-3f697d9a824ebfcfee56f875c933a732_l3.png)
![Rendered by QuickLaTeX.com 0.6p = 3p - 0.8](https://theculture.sg/wp-content/ql-cache/quicklatex.com-fe64b9d7bb65e7214e8701b13b0e8014_l3.png)
![Rendered by QuickLaTeX.com p = \frac{1}{3}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-e3363b49ce3b9637e3e0adbbf729b81f_l3.png)
(ii)
is the probability of event B occurring and event A not occurring.
![Rendered by QuickLaTeX.com \text{P}(A' \cap B) = \text{P}(A \cup B) - \text{P}(A)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ad3736285443f78aaa128065a79de94b_l3.png)
![Rendered by QuickLaTeX.com \text{P}(A' \cap B) = 0.8 - \frac{1}{3} = \frac{7}{15}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-da8b24b966d9ea63877b09114ca7b040_l3.png)
(i)
Required Probability
![Rendered by QuickLaTeX.com = \frac{112}{300} = \frac{28}{75}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-aa440291f967035d45b990ec138e19cb_l3.png)
(ii)
Required Probability ![Rendered by QuickLaTeX.com = \frac{140}{300} + \frac{112}{300} - \frac{40}{300} = \frac{53}{75}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-9befb703a9e66ef668b43112a675aebf_l3.png)
(iii)
Required Probability ![Rendered by QuickLaTeX.com = \frac{54}{300} = 0.18](https://theculture.sg/wp-content/ql-cache/quicklatex.com-30d347ddeb9778c8f0c0da03232789c4_l3.png)
(iv)
![Rendered by QuickLaTeX.com \text{P}(M) \times \text{P}(A) = \frac{160}{300} \times \frac{90}{300} = 0.16 \neq \text{P}(M \cap A)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-187028456068e8562d30d0e38ee0b211_l3.png)
Thus, events
and
are not independent events.
(v)
Required Probability
(3sf)
(i)
[Graph]
(ii)
Using GC,
(3sf)
The product moment correlation coefficient shows a strong negative linear correlation between the times to run 100 metres and the times to run 10000 metres.
(iii)
Using GC,
(2dp)
(iv)
(3sf)
Thus, required time taken is 37.2 minutes.
(v)
Since
is within the data range, the estimate is reliable as we are performing interpolation.
(i)
Unbiased estimate of population mean
![Rendered by QuickLaTeX.com = \frac{15}{100} + 30 = 30.15](https://theculture.sg/wp-content/ql-cache/quicklatex.com-3e7debace1a00b0f80db447f42692202_l3.png)
Unbiased estimate of population variance ![Rendered by QuickLaTeX.com = \frac{1}{99}(82 - \frac{15^2}{100}) = \frac{29}{36}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0ce305408e9e9da2fefc0271d3cd28d2_l3.png)
(ii)
Let
denote the population mean length of the fish.
![Rendered by QuickLaTeX.com H_0: \mu = 30](https://theculture.sg/wp-content/ql-cache/quicklatex.com-091a85576992fb0646e966370b3c4804_l3.png)
![Rendered by QuickLaTeX.com H_1: \mu > 30](https://theculture.sg/wp-content/ql-cache/quicklatex.com-c2da2123695b17878c812e444482186f_l3.png)
Under
approximately by Central Limit Theorem, since the sample size is sufficiently large.
We perform a one tailed Z test at 2.5% level of significance.
Using GC, ![Rendered by QuickLaTeX.com p = 0.047335](https://theculture.sg/wp-content/ql-cache/quicklatex.com-b7fe697845a0dcbe2ee302f888b427e1_l3.png)
Since
, we reject the scientist’s claim at 2.5% of significance level, and conclude with insufficient evidence that the mean length of fishes is greater than 30 cm.
(iii)
It is not necessary to assume as the sample size of 100 is sufficiently large for us to approximate the sample mean lengths of fishes to a normal distribution.
(iv)
![Rendered by QuickLaTeX.com H_0: \mu = 30](https://theculture.sg/wp-content/ql-cache/quicklatex.com-091a85576992fb0646e966370b3c4804_l3.png)
![Rendered by QuickLaTeX.com H_1: \mu > 30](https://theculture.sg/wp-content/ql-cache/quicklatex.com-c2da2123695b17878c812e444482186f_l3.png)
Under
approximately by Central Limit Theorem, since the sample size is sufficiently large.
To reject
at 10% of significance level,
![Rendered by QuickLaTeX.com \frac{m - 30}{\sqrt{\frac{0.9}{100}}} > 1.281551567](https://theculture.sg/wp-content/ql-cache/quicklatex.com-2249a8df0542729b2cbb99e1992d1a53_l3.png)
![Rendered by QuickLaTeX.com m > 30.12157866](https://theculture.sg/wp-content/ql-cache/quicklatex.com-705b0f31bc664b4f45f6b92ab52afc88_l3.png)
(round up to 3sf)
Note: round up here to avoid for eg,
which will result in non-rejection of claim.
Let
![Rendered by QuickLaTeX.com A](https://theculture.sg/wp-content/ql-cache/quicklatex.com-783721c5f28851887dd4199f225c37be_l3.png)
and
![Rendered by QuickLaTeX.com B](https://theculture.sg/wp-content/ql-cache/quicklatex.com-501c081517b1f5e359fed99657131700_l3.png)
denote the masses of Type A and Type B components respectively.
![Rendered by QuickLaTeX.com A \sim \text{N}(250, 3^2)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-56156619e8dd87c27be2e59385e8aa9d_l3.png)
![Rendered by QuickLaTeX.com B \sim \text{N}(240, 4^2)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-e4891f717b57ef1499c40cf399b79504_l3.png)
(i)
(3sf)
(ii)
![Rendered by QuickLaTeX.com A - B \sim \text{N}(10, 25)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-8d4edc3c6756a6093c1454f2687e1df5_l3.png)
(3sf)
(iii)
![Rendered by QuickLaTeX.com A_1 + \ldots A_6 + B_1 + B_2 + B_3 \sim \text{N}(2220, 102)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-4846c48ba74866fef77899b6b63abb4f_l3.png)
(3sf)
(iv)
Let
and
denote the cost of producing a Type A and Type B components respectively.
![Rendered by QuickLaTeX.com X \sim \text{N}(5, 0.0036)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-d2fb48a5e3734e0bfa2bbb4d4b8f5c1d_l3.png)
![Rendered by QuickLaTeX.com Y \sim \text{N}(7.2, 0.0144)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-e4a6bc5f05d79ce7223d7a43ed58deb2_l3.png)
![Rendered by QuickLaTeX.com Y_1 + \ldots +Y_{10} - (X_1 + \ldots + X_{10}) \sim \text{N}(22, 0.18)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-079be9f7f34fcce07032bbcf1cf36ecb_l3.png)
(3sf)