Question

thickness measurements of ancient prehistoric native american pot shards discovered in a hopi village were approximately normally distributed with a mean of 5.1 mm and a standard deviation of 0.9 mm. for a randomly found shard, what is the probability that the thickness is

(a) less than 3.0 mm

(b) more than 7.0 mm

(c) between 3.0 and 7.0 mm

Answer 1

mean = 5.1mm; stdev = 0.9mm

Z= (xbar - mu)/stdev

a) Probability that the thickness is less than 3.0 mm is given by the area under the Z curve on left hand side of the point xbar = 3

=P[Z = (3 - 5.1)/0.9] = P[Z = -2.33] = 0.0099

b) Probability that the thickness is more than 7.0 mm is given by the area under the Z curve on right hand side of the point xbar = 7

=1 - P[Z = (7 - 5.1)/0.9] =1 - P[Z = 2.11] = 1 - 0.9826 = 0.0174

We have subtracted from 1 since this is a right tailed statistic.

P[X <= a] + P[X > a] = 1

The Z curve gives us the probability upto the point X bar i.e. (X < xbar). SO in order to calculate the area (X > xbar) we subtract it from 1

c) Probability that the thickness is between 3.0 and 7.0 mm is given by the area under the Z curve between the points xbar = 3 and 7

= P[Z = (7 - 5.1)/0.9] - P[Z = (3 - 5.1)/0.9]

= P[Z = 2.11] - P[Z = -2.33]

= 0.9826 - 0.0099

= 0.9727