Question

Every year at the AP Statistics Exam reading apparel is sold to commemorate the event. Based on historical data, we know that 56% of teachers will make an apparel purchase. This year, the organizers have decided to select a random sample of 50 teachers. If a selected teacher made an apparel purchase, they will be given a prize. What is the probability that more than 60% of the selected teachers will win a prize

Answer 1

Answer:

0.284

Step-by-step explanation:

To carry out this calculation, we begin by describing the sampling distribution of the sample proportion.

The sample size is n = 50 and the population proportion of teachers who made an apparel purchase is 0.56.

Shape: Because np = (50)(0.56) = 28 and n(1 – p) = (50)(0.44) = 22 are both at least 10, the shape of the sampling distribution of the sample proportion is approximately Normal.

Center:

μ

p

^

=

p

=

0

.

5

6

μ

p

^

​

​ =p=0.56

Variability: The standard deviation of the sample proportion is approximately

(

0

.

5

6

)

(

0

.

4

4

)

5

0

≈

0

.

0

7

0

2

50

(0.56)(0.44)

​

​ ≈0.0702.

P(

p

^

p

^

​ > 0.6) = Normalcdf(lower: 0.6, upper: 1000, mean: 0.56, SD: 0.0702) = 0.284.

P

(

p

^

>

0

.

6

)

=

P

(

z

>

0

.

6

−

0

.

5

6

0

.

0

7

0

2

)

=

P

(

z

>

0

.

5

7

)

=

1

−

0

.

7

1

5

7

=

0

.

2

8

4

3

P(

p

^

​ >0.6)=P(z>

0.0702

0.6−0.56

​ )=P(z>0.57)=1−0.7157=0.2843