<ol class="i-list"><li><div class="list-item-paragraph">Factory $A$ produces nuts whose mass may be assumed to be normally distributed with mean $\mu $ grams and standard 	deviation $\sigma $ grams. A random sample of $50$ nuts is taken. It is given that the probability that the mean mass is less than $247$ grams is $0.018079$, correct to $5$ significant figures. It is also given that the probability that the total mass exceeds $12600$ grams is $0.78397$, correct to $5$ significant figures. Find the values of $\mu $ and $\sigma $, giving your answers to the nearest grams. <span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">(For this question, you should state clearly the values of the parameters of any normal distribution you use.) <br>Factory $B$ produces bolts and nuts. The masses, in grams, of bolts and nuts produced are modelled as having independent normal distributions with means and standard deviations as shown in the table: <br></div><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 50px;"><col style="width: 200px;"><col style="width: 200px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Mean Mass (in grams)</strong></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Standard Deviation (in grams)</strong></p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Bolts</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$745$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$7.3$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Nuts</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$250$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></td></tr></tbody></table></div><ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that the mass of a randomly chosen bolt differs from $3$ times the mass of a randomly chosen nut by at least $40$ grams. 	<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">This factory introduces a new process which is able to reduce the mass of each nut by $10%$. Find the probability that the total mass, after the introduction of this process, of $10$ randomly chosen nuts is less than $2.24$ kg. 	<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Factory $A$ produces nuts whose mass may be assumed to be normally distributed with mean $\mu $ grams and standard 	deviation $\sigma $ grams. A random sample of $50$ nuts is taken. It is given that the probability that the mean mass is less than $247$ grams is $0.018079$, correct to $5$ significant figures. It is also given that the probability that the total mass exceeds $12600$ grams is $0.78397$, correct to $5$ significant figures. Find the values of $\mu $ and $\sigma $, giving your answers to the nearest grams. 

(For this question, you should state clearly the values of the parameters of any normal distribution you use.) 
Factory $B$ produces bolts and nuts. The masses, in grams, of bolts and nuts produced are modelled as having independent normal distributions with means and standard deviations as shown in the table: 


Sampling

International Baccalaureate Higher Level Mathematics

2017 MJC P2 Q10

2017 MJC P2 Q10