Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution

Question

Use Ti-84

1) In one country, the conviction rate for speeding is 85%. Estimate the probability that of the next 100 speeding summons issued there will be at least 90 convictions.

2) The probabilty that a radish seed will germinate is 0.7. Estimate the probability that of 140 randomly selected seeds, exactly 100 will germinate.

Answer 1

Step 1:

(1)

The conviction rate of speeding is 85 %.

Sample size n is 100.

Probability of success p is 85% = 0.85.

Probability of failure q is 1-0.85 = 0.15.

Calculate mean.

Mean .

.

Calculate standard deviation.

Standard deviation .

.

Step 2:

Find the probability that atleast there are 90 convictions .

Now calculate the Z-score for the given probability.

,

where x is the conviction rate for speeding summons,

is the mean and

is the standard deviation.

For x = 90, Z-score with correction factor (90-0.5) is .

From the Z-score table, Area of the region when Z = 1.260 is 0.8962.

The probability of conviction rate for speeding summons greater than or equal to 90 is 0.0808.

Check:

Using TI-84 Calculator.

Follow these steps to evaluate .

1.Select normalcdf() 

[2nd --> VARS --> 2 ]

2.Enter the values of mean, standard deviation, limit.

normalcdf(89.5, 9999, 85, 3.571)

Note: The 9999 is used as the highest value since we need to calculate .

3.Now press Enter in calculator to view answer

normalcdf(89.5, 9999, 85, 3.571)

=0.1038

Solution:

The probability of conviction rate for speeding summons is 0.1038.

Answer 2

Step 1:

(2)

The probability that a radish seed will germinate is 0.7.

Number of trails n is 140.

Find the probability of exactly 100 seeds will germinate .

Binomial Probability ,

where n is the Number of trails.

p is the probability of success on a single trial.

r is the number of success trail.

The probability of exactly 100 seeds will germinate is 0.06952.

Check:

Using TI-84 Calculator.

Follow these steps to evaluate .

1.Select binomialpdf() 

[2nd --> VARS --> scroll down --> 0 ]

2.Enter the values of sample size, probability and x.

binomialpdf(140, 0.7, 100)

3.Now press Enter in calculator to view answer

binomialpdf(140, 0.7, 100)

= 0.06952

Solution:

The probability of exactly 100 seeds will germinate is 0.06952.