Find the probability that a bag of rice chosen at random weighs between 990 g and 1004 g.

95% of the bags of rice weigh less than $k$ grams.

Find the value of $k$.

For a bag of rice chosen at random, ${\text{P}}(1000 - a < W < 1000 + a) = 0.9$.

Find the value of $a$.

     (M1)

Note: Award (M1) for approximate curve with 990 and 1004 in correct place.

$0.835\;\;\;(0.835135 \ldots ,{\text{ }}83.5\% )$     (A1)     (C2)

     (M1) 

Note: Award (M1) for approximate curve with $k$ placed to the right of the mean.

$1010\;\;\;(1006.57 \ldots )$     (A1)     (C2)

Note: Award full marks only for $1010$, $1007$ or an answer with more than $4$ sf resulting from correct rounding of $1006.57...$.

     (M1)

Note: Award (M1) for some indication of symmetry on diagram.

OR

${\text{P}}(W < 1000 - a) = 0.05\;\;\;$OR$\;\;\;{\text{P}}(W > 1000 + a) = 0.05$     (M1)

Note: Award (M1) for probability with single inequality resulting from symmetry of diagram.

$(a = ){\text{ }}6.58\;\;\;(6.57941 \ldots )$     (A1)     (C2)

A significant number of candidates did not answer this question. It was very rare that a correct method was shown for any of the parts of this question. Often a normal distribution graph was drawn with indication of the mean and multiples of the standard deviation, with indication of the corresponding probabilities, but not a diagram identifying the area under the curve corresponding to the questions. There were however many correct answers for part (a). For part (b) many answered incorrectly; the most common incorrect answer was 1008, resulting from adding 2 sd to the mean. Very few correct answers were given for part (c).

A significant number of candidates did not answer this question. It was very rare that a correct method was shown for any of the parts of this question. Often a normal distribution graph was drawn with indication of the mean and multiples of the standard deviation, with indication of the corresponding probabilities, but not a diagram identifying the area under the curve corresponding to the questions. There were however many correct answers for part (a). For part (b) many answered incorrectly; the most common incorrect answer was 1008, resulting from adding 2 sd to the mean. Very few correct answers were given for part (c).

A significant number of candidates did not answer this question. It was very rare that a correct method was shown for any of the parts of this question. Often a normal distribution graph was drawn with indication of the mean and multiples of the standard deviation, with indication of the corresponding probabilities, but not a diagram identifying the area under the curve corresponding to the questions. There were however many correct answers for part (a). For part (b) many answered incorrectly; the most common incorrect answer was 1008, resulting from adding 2 sd to the mean. Very few correct answers were given for part (c).