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Date May 2018 Marks available 2 Reference code 18M.2.sl.TZ2.10
Level SL only Paper 2 Time zone TZ2
Command term Find Question number 10 Adapted from N/A

Question

The mass M of apples in grams is normally distributed with mean μ. The following table shows probabilities for values of M.

The apples are packed in bags of ten.

Any apples with a mass less than 95 g are classified as small.

Write down the value of k.

[2]
a.i.

Show that μ = 106.

[2]
a.ii.

Find P(M < 95) .

[5]
b.

Find the probability that a bag of apples selected at random contains at most one small apple.

[3]
c.

Find the expected number of bags in this crate that contain at most one small apple.

[3]
d.i.

Find the probability that at least 48 bags in this crate contain at most one small apple.

[2]
d.ii.

Markscheme

evidence of using \(\sum {{p_i}}  = 1\)     (M1)

eg   k + 0.98 + 0.01 = 1

k = 0.01     A1 N2

[2 marks]

a.i.

recognizing that 93 and 119 are symmetrical about μ       (M1)

eg   μ is midpoint of 93 and 119

correct working to find μ       A1

\(\frac{{119 + 93}}{2}\)

μ = 106     AG N0

[2 marks]

a.ii.

finding standardized value for 93 or 119      (A1)
eg   z = −2.32634, z = 2.32634

correct substitution using their z value      (A1)
eg  \(\frac{{93 - 106}}{\sigma } =  - 2.32634,\,\,\frac{{119 - 106}}{{2.32634}} = \sigma \)

σ = 5.58815     (A1)

0.024508

P(X < 95) = 0.0245      A2 N3

[5 marks]

b.

evidence of recognizing binomial    (M1) 

eg 10, ananaCpqn−=××and 0.024B(5,,)pnp= 

valid approach    (M1) 

eg P(1),P(0)P(1)XXX≤=+= 

0.976285 

0.976     A1 N2 

[3 marks]

c.

recognizing new binomial probability      (M1)
eg     B(50, 0.976)

correct substitution      (A1)
eg     E(X) = 50 (0.976285)

48.81425

48.8    A1 N2

[3 marks]

d.i.

valid approach      (M1)

eg   P(X ≥ 48), 1 − P(X ≤ 47)

0.884688

0.885       A1 N2

[2 marks]

d.ii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.
[N/A]
d.i.
[N/A]
d.ii.

Syllabus sections

Topic 5 - Statistics and probability » 5.9 » Normal distributions and curves.
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