1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
vladimir2022 [97]
3 years ago
15

A coin is thrown independently 10 times to test the hypothesis that the probability of heads is 0.5 versus the alternative that

the probability is not 0.5. The test rejects the null hypothesis if either 0 or 10 heads are observed.
(a) What is the significance level of the test?
(b) If, in fact, the probability of heads is 0.1, what is the power of the test?
Mathematics
1 answer:
mafiozo [28]3 years ago
3 0

Answer:

(a) The significance level of the test is 0.002.

(b) The power of the test is 0.3487.

Step-by-step explanation:

We are given that a coin is thrown independently 10 times to test the hypothesis that the probability of heads is 0.5 versus the alternative that the probability is not 0.5.

The test rejects the null hypothesis if either 0 or 10 heads are observed.

Let p = <u><em>probability of obtaining head.</em></u>

So, Null Hypothesis, H_0 : p = 0.5

Alternate Hypothesis, H_A : p \neq 0.5

(a) The significance level of the test which is represented by \alpha is the probability of Type I error.

Type I error states the probability of rejecting the null hypothesis given the fact that the null hypothesis is true.

Here, the probability of rejecting the null hypothesis means we obtain the probability of observing either 0 or 10 heads, that is;

            P(Type I error) = \alpha

         P(X = 0/H_0 is true) + P(X = 10/H_0 is true) = \alpha

Also, the event of obtaining heads when a coin is thrown 10 times can be considered as a binomial experiment.

So, X ~ Binom(n = 10, p = 0.5)

P(X = 0/H_0 is true) + P(X = 10/H_0 is true) = \alpha

\binom{10}{0}\times 0.5^{0} \times (1-0.5)^{10-0}  +\binom{10}{10}\times 0.5^{10} \times (1-0.5)^{10-10}  = \alpha

(1\times 1\times 0.5^{10})  +(1 \times 0.5^{10} \times 0.5^{0}) = \alpha

\alpha = 0.0019

So, the significance level of the test is 0.002.

(b) It is stated that the probability of heads is 0.1, and we have to find the power of the test.

Here the Type II error is used which states the probability of accepting the null hypothesis given the fact that the null hypothesis is false.

Also, the power of the test is represented by (1 - \beta).

So, here, X ~ Binom(n = 10, p = 0.1)

1-\beta = P(X = 0/H_0 is true) + P(X = 10/H_0 is true)

1-\beta = \binom{10}{0}\times 0.1^{0} \times (1-0.1)^{10-0}  +\binom{10}{10}\times 0.1^{10} \times (1-0.1)^{10-10}  

1-\beta = (1\times 1\times 0.9^{10})  +(1 \times 0.1^{10} \times 0.9^{0})

1-\beta = 0.3487

Hence, the power of the test is 0.3487.

You might be interested in
A triangular bandana has an area of 74 square inches. The height of the triangle is 9 1 4 inches. Enter and solve an equation to
Lilit [14]
Hi there!

The formula for the area of a triangle is A = 1/2(b x h). Using the information provided, all we have to do is plug in what we know and solve!

WORK:
74 = 1/2(b x 9.14)
148 = b x 9.14
b = 16.2 inches (approximately)

Hope this helps!! :)
6 0
3 years ago
A vendor has 12 balloons for sale: 6 are yellow, 2 are red, and 4 are green. A balloon is selected at random and sold. If the ba
olchik [2.2K]
Answer is 5 over 11. I hope this may help you
8 0
3 years ago
Read 2 more answers
4.
natali 33 [55]
The answer is D. 15+18+12+5= 50 and there are 5 ramblers
3 0
2 years ago
Read 2 more answers
a taxi service charges an initial fee of $5 plus $1.80 per mile. How many miles can you travel for $12
user100 [1]
First you need to write an equation or an inequality.

5+1.80x<=12 
the x here is how many miles you are paying for, now you just solve for x. 

5+1.80x<=12
-5                 -5

1.80x<=7
/1.80         /1.80
x<=3.8

you can travel only three whole miles, you do not have enough money to go four full miles.
4 0
3 years ago
WILL MARK BRAINIEST!!!
Zarrin [17]
R^2/2(pi/180*D-sin(D)) is the formula so 6^2/2(pi/180*120-sin(120)) is the problem which comes to about 22cm^2
4 0
2 years ago
Read 2 more answers
Other questions:
  • Cos 52º=______
    14·2 answers
  • 16. The length of the hypotenuse of a 45°-
    5·1 answer
  • A. 5y2 - 60 = 0<br>by Square root property​
    13·1 answer
  • What multiplied by 3 equals 15 but added by 3 equals 18
    10·2 answers
  • PLEAASEEE HELPPPP!!!!!!
    8·1 answer
  • What does f(3)= ? <br> Please help answer me
    10·1 answer
  • 2. Explain, in words, how you would solve this problem. Hint: you should get 1 point
    7·2 answers
  • What would you expect to happen if a pay rate of $13.50 per hour were offered?
    14·1 answer
  • Someone help with this please?
    10·1 answer
  • Evaluate: m2•m3 helppppppppppppppoo
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!