Answer to Question #146654 in Statistics and Probability for Cadbyry

Question

“Chief Grill” is a local restaurant, its annual data shows that the average sale of chicken steaks is 1200 serving per month with a standard deviation of 102. If the data is normally distributed; calculate the probability of sale of chicken steaks in the next month will be:
a. less than 1000 servings
b. more than 1400 servings
c. between 1100 to 1300 serving
d. less than 1000 but not less than 1350

Answer 1

X ~ P(M, σ²)

M = 1200

σ = 102

a. P(less than 1000 servings) = P(X < 1000)

$= P(\frac{X-M}{σ} < \frac{1000-M}{σ})$

$= P(Z < \frac{1000-1200}{102})$

=P(Z < -1.96) = 0.025

b. P(more than 1400 servings) = P(X>1400)

$= P(\frac{X-M}{σ} > \frac{1400-M}{σ})$

$= P(Z > \frac{1400-1200}{102})$

= P(Z > 1.96)

= 1 – P(Z ≤ 1.96) = 1 – 0.975 = 0.025

c. P(between 1100 to 1300 serving) = P(1100 < X < 1300)

$= P(\frac{1100 – M}{σ} < \frac{X – M}{σ} < \frac{1300 – M}{σ})$

$= P(\frac{1100 – 1200}{102} < Z < \frac{1300 – 1200}{102})$

= P(-0.98 < Z < 0.98)

=P(Z ≤ 0.98) – P(Z ≤ -0.98) = 0.8365 – 0.1635 = 0.673

d. P(less than 1000 but not less than 1350) = P(1000 > X) + P(X> 1350)

$= P(\frac{1000 – M}{σ} > \frac{X – M}{σ}) + P(\frac{X – M}{σ} > \frac{1350 – M}{σ})$

$= P(\frac{1000 – 1200}{102} > Z) + P(Z > \frac{1350 – 1200}{102})$

= P(-1.96 > Z) + P(Z > 1.47)

= P(Z < -1.96) + (1 – P(Z ≤ 1.47)) = 0.025 + (1 – 0.9292) = 0.0958