(a)
0.115

\( X \sim N(4, 0.25^2) \)

Seek: \( P(X < 3.7) \)

Method 1
Z-score: \( z = \frac{3.7 – 4}{0.25} = -1.2 \)

\( P(Z < -1.2) = 1 – P(Z < 1.2) \approx 1 – 0.8849 = 0.1151 \)

Round to 3 decimal places: 0.115

Method 2
Visualize normal curve, vertical line at 3.7 cm, left of mean, shade left region


Corresponds to \( P(X < 3.7) \), \( z = -1.2 \)

\( P(Z < -1.2) \approx 0.1151 \approx 0.115 \)

Result: 0.115 [2]

(b)
4.13

Seek: \( P(X > k) = 0.3 \), so \( P(X \leq k) = 0.7 \)

Method 1
\( P(Z \leq z) = 0.7 \), \( z \approx 0.524 \)

Z-score: \( \frac{k – 4}{0.25} = 0.524 \)

\( k – 4 = 0.524 \times 0.25 = 0.131 \)

\( k = 4 + 0.131 = 4.131 \)

Round to 2 decimal places: 4.13

Method 2
Visualize normal curve, vertical line at \( k \) cm, right of mean, shade right for 0.3


\( P(X \leq k) = 0.7 \), \( z \approx 0.524 \)

\( k \approx 4.131 \approx 4.13 \)

Result: 4.13 [2]

(c)
0.210

Seek: \( P(4 – m < d < 4 + m) = 0.6 \)

Method 1
Symmetric interval: \( P(|Z| < \frac{m}{0.25}) = 0.6 \)

\( 2P(Z < \frac{m}{0.25}) – 1 = 0.6 \)

\( P(Z < \frac{m}{0.25}) = 0.8 \), \( z \approx 0.8416 \)

\( \frac{m}{0.25} \approx 0.8416 \)

\( m \approx 0.8416 \times 0.25 = 0.2104 \)

Round to 3 decimal places: 0.210

Method 2
Visualize normal curve, symmetric lines at \( 4 – m \) and \( 4 + m \), shade between for 0.6


\( P(Z < \frac{m}{0.25}) = 0.8 \), \( z \approx 0.8416 \)

\( m \approx 0.2104 \approx 0.210 \)

Result: 0.210 [2]