Statistics and Probability Answers

1.A fitness center claims that its members lose an average of 12 pounds or more the first month after joining the center. An independent agency that wanted to check this claim took a sample of 45 members and found that they lost an average of 10 pounds within the first month with standard deviation of 3 pounds. Find the p-value for this test. What will your decision be if 𝛼 = 0.05?

Carl has completed 8 math homework assignments that are all worth 20 points, and his mean score is 17. Carlos has two more 20-point homework assignments to complete before his progress report gets sent out, and he wants to raise his mean score to an 18. Is it possible for Carlos to raise his mean score to an 18? If it is possible, explain what scores he would need to get, and if not, explain why not.

Suppose that the probability that a corn seed from a certain batch does not germinate equals 0.02. If we plant 200 of these seeds, what is the probability that

• A) At most 5 seeds will not germinate?

• B) Exactly 3 will not germinate?

• C) At least 3 will not germinate?

An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil.

• A) What is the probability that the first strike comes on the third well

drilled?

• B) What is the probability that the third strike comes on the seventh

well drilled?

A box of bulbs is inspected. The inspector draws a bulb randomly and independently, and tests it until he finds 10 defective bulbs. Suppose 5% of all bulbs are defective. Let X be the number of bulbs tested in order to find 10 defective bulbs.

• A) What is the distribution of X?

• B) What is the expected number of bulbs that must be tested in order to get 10 defective bulbs?

• C) What is the variance of X?

• D) What is the probability that 15 bulbs will be tested in order to find

10 defective bulbs?

1.Let the random variable X be the total number of heads when a fair coin is tossed 4 times.

•Is this a Bernoulli or binomial trial/experiment?

•Are the conditions of a Binomial trial satisfied?

•Find the probability mass function for X.

2. The probability that the patient recovers from a rare blood disease is 0.4. If 10 people are known to have contracted the disease, what is the probability that

a) exactly 3 patients survive the disease?

b) from 3 to 5 patients survive the disease?

3. Suppose that a student is given a test with 10 true-false questions. Let X be the number of questions guessed correctly by the student.

i) Write down the probability distribution (probability mass function) of X

ii) What is the probability that the student guesses all questions correctly?

iii) what is the probability that is the probability that the student guesses 7 or more questions

correctly?

FIND THE MEAN OF THE SET OF DATA BELOW AND CONSTRUCT A SAMPLING DISTRIBUTION

BY SELECTING 3 SAMPLES AT A TIME: 7 10 14 17 20

The distribution function of a discrete random variable X is given by

F_x(k) = "\\begin{Bmatrix}\n 0, & k < 1 \\\\\n 0.1, & 1 \\leq k <2 \\\\\n 0.3, & 2 \\leq k <3\\\\\n0.7, & 3 \\leq k <4 \\\\\n0.8 & 4 \\leq k < 5\\\\\n1 & k \\geq 5\n\\end{Bmatrix}"

1) What are the possible values of X?

2) Find the probability mass function of X.

Suppose that 4 children are planned in a family. Let the random variable X stand for the number of boys. Discuss the probability distribution of X.

• Find the mean and variance of the random variable X.

Consider the experiment of throwing a pair of dice. We are interested in the sum of the two numbers that occur. We know that X can assume the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

1) What is the probability mass function of X?

2) What is the probability that the number of dots will be at least 4 and at most 7?

3) Find the probability that the sum will be at least 7.