Question

The monthly demand for a portable solar lights is distributed in a normal way with μ = 1,200 units and σ = 100 units. Determine the probability that in December sales: 1. Exceed 1,000 units? 2. Are there between 1,100 and 1,300 units? 3. Are they below 900 units? 4. Are more than 1,800 units? 5. Are less than 700 units? 6. How many units of the product should you have in inventory to ensure What satisfies sales and does not fall short?

Answer 1

μ =1200

σ = 100

Z = X - μ /σ

1) P( X > 1000)

= P( z > 1000 - 1200 / 100)

= P( z > -2)

= 0.4772 + 0.5 [standard normal distribution table]

= 0.9772

2) P(1100 < X < 1300)

= P( 1100 -1200 / 100 < z < 1300 - 1200 /100)

= P( -1 < z < 1)

= 0.3413 + 0.3413 [standard normal distribution table]

= 0.6826

3)

P( X < 900)

= P( z < 900 - 1200 /100)

= P( z < -3)

= 0.5 - 0.4987. [standard normal distribution table]

= 0.0013

4)

P( X > 1800)

= P( z > 1800 -1200/100)

= P( z > 6)

= 0 (approx.)

5)

P( x < 700)

= P( z < 700 -1200/100)

= P( z < -5 )

= 0 (approx.)

6)

Number of units should have in inventory to satisfies sales and does not fall short means product should have under 99.9% confidence interval

From standard normal distribution table,

At 99.9% confidence , maximum z value = + 3.09

Z = X - μ / σ

3.09 = X - 1200 /100

X = 1200 + 309

X = 1509

So, inventory should have 1509 units, which satisfies sales and does not fall short.