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Question

1) The lengths of pregnacies of are normally distributed with a mean of 268 days and a standard deviation of 15 days.

a) One classical use of the normal distribution is inspired by a letter to "Dear Abby" in which a wife claimed to have given birth 308 days after a brief visit from her husband, who was serving in the Navy. Given this information, find the probability of a pregnancy lasting 308 days or longer?

b) If we stipulate that a baby is premature if the length of pregnacy is in the lowest 3%, find the length that separates premature babies from those are not premature.

2) A Richter scale magnitudes of earthquakes are normally distributed with a mean of 1.184 and a standard deviation of 0.587.

a) Earthquakes with magnitues less than 2.000 are considred "microearthquakes" that are not felt. What percentage of earthquakes fall into this category?

b) Earthquakes above 4.0 will cause indoor items to shake. What percentage of earthquakes fall into this category?

c) Find the 95th percentile. Will all earthquakes above the 95th precentile. Will all earthquakes above the 95th percentile cause indoor items to shake?

Answer 1

Step 1:

1 (a)

Let the random variable  be the length of pregnancy.

The mean of random variable is 268 days.

The standard deviation of random variable is 15 days.

Find the probability of a pregnancy lasting 308 days or longer.

.

Now calculate the Z-score for the given probability.

,

where is the length of pregnancy,

is the mean and

is the standard deviation.

For , Z-score is

From the Z-score table, Area of the region when Z = 2.67 is 0.9962.

The probability of a pregnancy lasting 308 days or longer is 0.0038.

Solution:

(a) The probability of a pregnancy lasting 308 days or longer is 0.0038.

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Answer 2

Step 1:

1 (b)

Find the length of the pregnancy .

The length of the pregnancy is in the lowest 3% = 0.03.

Area of the region is 0.03

From Z-score table, Z-score when area of the region is 0.03 is .

Now calculate length of the pregnancy .

Z-score

The length of the pregnancy is 240 days.

Solution:

(b) The length of the pregnancy is 240 days.

Answer 3

(2)

Step 1:

(a)

Let the magnitude of earthquake is .

The mean of earthquake is 1.184.

The standard deviation of earthquake is 0.587.

Find the percentage of earthquake with magnitudes less than 2.000.

Percentage of earthquake with magnitudes less than 2 is .

Now calculate the Z-score for the given probability.

,

where  is the magnitude of earthquake,

 is the mean and

 is the standard deviation.

From z - score table the percentage is  .

Solution:

The percentage of earthquake with magnitudes less than 2 is 91.7%.

Answer 4

(2)

Step 1:

(b)

Let the magnitude of earthquake is .

The mean of earthquake is 1.184.

The standard deviation of earthquake is 0.587.

Find the percentage of earthquake with magnitudes above 4.

.

Now calculate the Z-score for the given probability.

,

where  is the magnitude of earthquake,

 is the mean and

 is the standard deviation.

Percentage of earthquake with magnitudes above 4 is .

From z - score table the percentage is  .

Percentage of earthquake with magnitudes above 4 is 0.00001%.

Solution:

Percentage of earthquake with magnitudes above 4 is 0.00001%.

Answer 5

(2)

Step 1:

(c)

Let the magnitude of earthquake is  .

The mean of earthquake is 1.184.

The standard deviation of earthquake is 0.587.

Find the 95th percentile of earthquake magnitudes.

From the z - score table :

z - score value corresponds to 95th percentile is 1.644.

Now calculate the earthquake magnitudes for the given percentile.

,

where  is the length of pregnancy,

 is the mean and

 is the standard deviation.

Magnitude of earthquake is 2.14.

Solution:

​Magnitude of earthquake is 2.14.