Probability and Statistics - Practice Tests and Solutions

A single die is rolled once. Find the probability of getting 1 or 6.

A ball is drawn at random from a box containing 20 red balls, 30 yellow balls, 10 green and 40 blue balls. Determine the probability that it is either green or blue.

If two dice are rolled, what is the probability of getting the sum of numbers greater than 7?

If X={1,3,5,7,9} , Y={1,2,3,9,11} and Z = {5,8,9}. What is the value of (X∩Y)∪(X ∩Z)?

If P(A) = 0.10, P(B) = 0.10, P(A ∪ B) = 0.18. What is the value of P(A∩B)?

Which of the following events are mutually exclusive?

If P(A) = 0.10, P(B) = 0.10, P(A∩B) = 0.02

P(C) = 0.10, P(A∩C) = 0.02, and events B and C are mutually exclusive event.

What is the value of P(B∩C)?

If P(A) = 0.10, P(B) = 0.10, P(A∩B) = 0.02

P(C) = 0.10, P(A∩C) = 0.02, and events B and C are mutually exclusive event.

What is the value of P(A∪B∪C)?

In a town, 50% of people speak English and 55% of people speak French. Out of these 13% people speak both English and French. What percent of people either speak English or French?

A card is drawn from a full deck of 52 cards. What is the probability that the card drawn will be either a king or a black card?

What is the probability of a 6 turning up at least once in two tosses of a fair die?

A fair die is tossed twice. What is the probability of getting the odd number in the first roll and even number in the second roll?

Two balls are drawn at random without replacement from a box containing 2 red balls, 3 yellow balls, 1 green and 4 blue balls. Determine the probability that both are red.

In how many ways can a committee of 4 people be chosen out of 10 people?

In how many different ways can the letters of the word ‘QUALITY' be arranged?

The probability that a random person has lung cancer is 0.0025 and the probability that the person has lung cancer and is also a heavy smoker is 0.002. Given that someone picked at random has lung cancer, what is the probability that the person is a heavy smoker?

If you randomly draw a card from two packs below and it comes out to be red.

What is the probability that the red card came from pack 1?

If P(A) = 0.17 and P(B) = 0.31. If P(A|B) = 0.12, what is P(B|A)?

A box contains 3 red and 2 blue balls while another box contains 2 red and 5 blue balls. A ball drawn at random from one of the boxes turns out to be red. What is the probability that it came from the first box?

A test for a rare disease is 99 percent correct most of the time (meaning if you have the disease, it will show that you do with a 99 percent probability, and if you do not have this disease, it will show that you don’t with a 99 percent probability).

The disease is very rare, and it occurs randomly in the population in one per 10,000 people.

If you get back the test results as positive, calculate the probability that you have the disease?

A welder produces welds that can have a crack, porosity or both. The probability of having crack is 0.10, and of having the porosity is 0.20.

What is the probability of having a weld with no defect?

A class has a mean score of 65 (μ=65) and a standard deviation of 7 (σ=7). Later 3 points are added to every student’s score. What are the new values for the mean and standard deviation?

In the birth register maintained by the hospital, one of the columns is the gender of the newborn child. What type of data is this?

What symbol is used to denote the mean of a population?

Find the variance of the following sample data:

1, 2, 3

If the standard deviation of the data is 0.36, what is the variance of this data?

The mean of 4 numbers is 28.

If three of the numbers are 10, 20, 40, what is the value of the fourth number?

What is the median of the following data set?

24, 4, 20, 8, 1, 17

What is the mode of the following data set?

24, 4, 20, 8, 1, 17, 4

What is the term used to describe the distribution of a data set that has 1 mode?

Which of the following measures can have more than one value for a set of data?

The mean and the standard deviation of two independent equal size groups are as follows: mean(A) = 100, sd(A) = 3, mean(B) = 25, sd(B) = 4. What will be the mean and standard deviation of (A-B) ?

Find the Inter-Quartile Range for the following data:

24, 4, 20, 8, 1, 17, 6

What is the mode of the data shown in the histogram below?

What is the median of the data shown in the Box-and-Whisker plot below?

What is the Inter-quartile Range of the data shown in the Box-and-Whisker plot below?

Which of the following statement is correct about the Box-and-Whisker Plot shown below?

The mean of a set of numbers is 100, the mode is 60, and the median is 80. What is the shape of the distribution?

Calculate the standard deviation of the following set of sample data:

1.1, 2.3, 4.0, 2.3, 1.7

A manufacturer produces 10% defective items. What is the probability that a sample of 4 random items picked by the client does not have a single defective?

A manufacturer produces 10% defective items. What is the probability that a sample of 4 random items picked by the client has one or more defectives?

What is the mean and the variance of a binomial random variable with n = 4 and p = 0.10?

In flipping a fair coin 5 times what is the probability of getting at least three heads?

The average defects rate of a supplier is 6%. In a simple random sample of six pieces by a client, what is the probability that there is at least one defective piece in that sample?

Calculate the probability of 3 or fewer defectives based on the following probabilities:

P(1 or more defective) = 0.83

P(2 or more defective) = 0.47

P(3 or more defective) = 0.17

P(4 or more defective) = 0.04

On a booking counter on the average 3.6 people come every 10 minute on weekends. You have been asked by your manager to find out the probability of getting more than 7 people in 10 minutes. What probability distribution would you use to solve this problem?

In the formula for the Poisson Distribution shown below, what is the value of “e”?

On a booking counter on the average 3.6 people come every 10 minute on weekends. What is the probability of getting exactly 7 people in 10 minutes?

A data entry operator has an average error rate of 0.1% of words typed. Consider the error rate follows the Poisson Distribution, what is the probability that an assignment of 2000 words will be error free?

What is the distribution that has the same mean and variance?

If the probability that the glass panel will have an air bubble (defect) is 0.001, what is the probability that out of 2000 panels produced none of the panels will have the air bubble? (Use Poisson Approximation)

A population has a μ=45 and σ=2. If these scores are transformed into z-scores, the population of z-scores will have a mean and standard deviation of:

A random variable X has a normal distribution, with a mean of 10 and a standard deviation of 2. What will be the z-score for a value of 5?

Using the Z Table what is the value of P(z < 1.13)?

Using the Z Table what is the value of P(–0.5 < z < 1.0)?

If data are normally distributed, what percentage of the data should lie within the range of mean plus/minus 3 times the standard deviation?

Looking at the below Histogram, what is the best estimate of the standard deviation of this distribution?

The mean weight of 1000 students at a certain college is 62 Kg and the standard deviation is 5Kg. Assuming that the weights are normally distributed, find the probability that a randomly selected student weighs between 55 and 60 Kg?

The average annual rain fall in a city is 35 inches. What is the standard deviation if 15% of the years have the rainfall above 40 inches? Assume yearly rainfalls are normally distributed.

Suppose that 40% of bolts have a tensile strength of more than 95 ksi, while 70% have more than 82 ksi. Assuming a normal distribution, what are the mean and standard deviation of the bolt tensile strength?

The lifetime of a newly produced LED bulb is normally distributed. The mean life is 14 years, and the standard deviation is 3 years. Out of 5,000 bulbs in test, how many are expected to fail in 5 years?

A fair coin is tossed 45 times. What is the probability that you will get heads in at most 25 of these tosses? (Use Normal approximation)

Regarding t-distribution which of the following statements is false?

A battery manufacturer claims that the battery lasts for 300 hours. An independent tester checks 15 batteries and find out the average life to be 280 hours with the standard deviation of 24 hours. What is the t-statistic in this example?

Bolts produced by a machine have a mean weight of 50 gm and a standard deviation of 2 gm. If 300 random samples of size 36 are drawn from this population, determine the expected mean and standard deviation of the sampling distribution of means.

From the Minitab output below, one item (SE Mean) has been blurred out. Calculate the missing value from the available data.

One thousand bolts produced by a machine have a mean weight of 50 gm and a standard deviation of 2 gm. What is the probability that a random sample of 100 bolts selected from this group will have a combined weight greater than 5,200 gm?

A Normal distribution has a mean of 50 and a standard deviation of 10.

If 100 items are randomly selected from this distribution, how many of these are expected to have value between 50 and 60?

Regardless of the distribution of the individuals, the distribution of the average of n samples will follow which distribution as n becomes large?

The distribution of a characteristic is negatively skewed. The sampling distribution of the mean for large samples is:

A survey was conducted in a country to determine the percentage of people who would support the change of government. The results were stated as 67% with a margin of error of ±4%. What is meant by ±4%?

Which of the following will result in the narrowest confidence interval?

Researchers want to determine the sleeping time each night in India. A study of a random sample of 100 Indians found the average amount of time people sleep each night is 6.3 hours with a standard deviation of 2.6 hours. Use the sample of data to construct a 95% confidence interval to estimate the true mean amount of time people in India sleep each night.

A teacher found that in a sample of 80 students, 17 said they use social media while doing their homework. Use the sample of data to construct a 90% confidence interval to estimate the true proportion of students using social media while doing their homework.

A medicine has a 66% success rate. The composition of the medicine was modified to improve its effectiveness. We want to test if with the new composition more than 66% get cured.

Which of the following is the correct null and alternate Hypothesis?

A lubricating oil manufacturing company continually monitors the viscosity of the oil. If the viscosity from sample data drops below a specified level, the production process is halted, and the machine is readjusted. Which of the following would result from a Type I error?

Suppose you conducted 10 hypothesis tests, each at the α = 0.05 significance level. What is the probability of committing a Type I error and incorrectly rejecting a true Ho with at least one of the 10 tests?

What is the probability of a Type II error when a hypothesis test is being conducted at the 10% significance level (α = 0.10)?

Which of the following statements is correct regarding the P-value?

What is the difference between setting the alpha value equal to 0.05 and alpha value equal to 0.01 in hypothesis tests?

The average breaking strength of steel rods is required to be at least 35,000 psi. Based on the historical information the standard deviation of breaking strength is 1,500 psi. A random sample of four specimens had the strength as piece one 32,000, piece two 36,000, piece three 34,000 and piece four 34,500.

Which of the following hypothesis tests would you conduct?

The average breaking strength of steel rods is required to be at least 35,000 psi. Based on the historical information the standard deviation of breaking strength is 1,500 psi. A random sample of four specimens had the strengths as: piece one 32,000, piece two 36,000, piece three 34,000 and piece four 34,500.

What will be null and alternate hypothesis is this case?

The average breaking strength of steel rods is required to be at least 35,000 psi. Based on the historical information the standard deviation of breaking strength is 1,500 psi. A random sample of four specimens had the strengths as: piece one 32,000, piece two 36,000, piece three 34,000 and piece four 34,500.

Calculate the test statistic.

The average breaking strength of steel rods is required to be at least 35,000 psi. Based on the historical information the standard deviation of breaking strength is 1,500 psi. A random sample of four specimens had the strengths as: piece one 32,000, piece two 36,000, piece three 34,000 and piece four 34,500.

What would you conclude from this test with a 95% confidence level?

A random sample of 15 batteries resulted in the average life of 280 hours with a standard deviation of 24 hours. Assume the battery life to be normally distributed and α = 0.05 test the following hypothesis:

Ho: ? = 300 hours| Ha: ? ≠ 300 hours

A survey claimed that 23% of adults in the country read a printed newspaper. A city newspaper does not agree with it and assumes that the percentage is more than 23% in the city. A city-wide survey was conducted. Out of 500 adults surveyed 124 people confirmed that they read the printed newspaper. Using the level of significance of 0.05 what would you conclude from this information?

To test the following hypothesis, what will be the test statistic?

Calculate the test statistic for the following hypothesis using the given data:

Conduct the following hypothesis using the given data. What would you conclude?

The chi-square test statistic can have the __________ .

A university has professors with Ph.D. and without a Ph.D. For comparing their average salaries, which of the following tests is most appropriate for a sample size of 10?

If alpha = 0.05, what do you conclude from the below output?

50 men and 50 women were surveyed to find out their annual income. The Surveyor wants to check if men have a significantly higher salary than women.

What will be the Null Hypothesis?

50 men and 50 women were surveyed to find out their annual income. The Surveyor wants to check if men have a significantly higher salary than women.

What will be the value of the test statistic?

In a two-sample t-test under what condition the following formula is used to calculate the degrees of freedom?

In a two-sample t-test under what condition the following formula is used to pool the standard deviation?

In a two-sample t-test, there are two approaches: consider the equal variance and considering the unequal variance. What test is used to check if the variance of the two populations is equal or not?

A two-sample t-test was conducted using Minitab. The value of df has been hidden in the output. What is the value of df in this case?

From vendor A we test 200 pieces and find 30 defectives. From vendor B we test 100 pieces and we find 10 defectives. To conduct two proportions test, what will be the value of pooled proportion?

From vendor A we test 200 pieces and find 30 defectives. From vendor B we test 100 pieces and we find 13 defectives. Conduct the two proportions test, to check if there is a significant difference in the quality of these two vendors. Use the Confidence level as 95%.

How is an F value calculated?

What attributes control the shape of an F distribution plot?

Use the tables to find F0.01 for an F random variable with df1=8 and df2=13.

Use the tables to find F0.95 for an F random variable with df1=13 and df2=8.

Two random samples taken from two normal populations yielded the following information. Find the F statistic.

Two random samples taken from two normal populations yielded the following information. An F Test was conducted with a 95% confidence level to check if the second sample has a higher variation. What will be the critical value of F?

Analysis of variance is a statistical method of comparing the ________ of several populations.

To test the equality of means of more than 2 populations which of the following techniques is used?

What does ANOVA calculate?

Three types of batteries were tested for battery life. See the battery lives in the number of hours in the table below.

In constructing the ANOVA table, what will be the values of factor and error degrees of freedom?

Find the missing values from the below one-way ANOVA table.

What is the value of Adj SS Total in the below one-way ANOVA table?

A fair coin is flipped 100 times, and the numbers of heads are counted. This experiment was repeated five times, and the resulted number of heads were: 51, 56, 54, 60 and 50. Is the coin biased? Check with a 95% confidence level.

The score of a test is normally distributed with a mean of 70 and a standard deviation of 10. The score of 100 random students is tabulated below. Find the calculated value of Chi-Square value for a Goodness-of-fit test.

Which of the following tests is used to analyze the results of a contingency table?

To test the effectiveness of a medicine, 235 patients were administered this medicine. The below table provides the gender vs effectiveness contingency table. What is the probability that it has a negative impact on a male?

To test the effectiveness of a medicine, 235 patients were administered this medicine. The below table provides the gender vs effectiveness contingency table. Is the drug effectiveness dependent upon gender? Test with a 95% confidence level.