Write Each Probability As Fractions In Simplest Form And R

Write Each Probability As Fractions In Simplest Form Andor Round Four

Write each probability as fractions in simplest form and/or round four decimal places. PART 1 - PROBABILITY I. Use the data set for gender and lied about your age A. Create a table from the data collected. B. Based on our sample, find the following probabilities 1. What is the probability that a randomly chosen Statistics student has lied about their age? 2. What is the probability that a randomly selected Statistics student has lied about their age given that the student is a female? 3. What is the probability that a randomly selected Statistics student has lied about their age given that the student is a male? 4. Determine whether the events of being female and lying about your age are independent or dependent. Explain your reasoning. PART 2 – DISCRETE PROBABILITY DISTRIBUTION II. Use the data set for number of pets. A. Construct and graph the discrete probability distribution from your data set. Note: You can use software (Excel, online sites, etc.) or draw them by hand using graph paper. B. Does your discrete probability distribution fulfill the two conditions it must satisfy? Explain. C. Find the mean, variance, and standard deviation of the distribution in part A above. Round to the nearest tenth. D. Find the following probabilities. 1. Using the probability distribution in part A, find the probability of randomly choosing a household that does not have a pet. 2. Using the probability distribution in part A, find the probability of randomly choosing a household that has more than 2 pets. PART 3 – BINOMIAL DISTRIBUTION III. Use the data set for the Registered Voters A. Find the probability that a randomly selected Statistics student is a registered voter. B. Construct and graph the binomial distribution for the number of voters when 10 students are selected. C. Find the mean, variance and standard deviation of this binomial distribution. Round to nearest tenth. D. If 10 Statistics students are selected at random, what is the probability that all of them are registered voters? E. If 10 Statistics students are selected at random, what is the probability that less than 5 of them are registered voters? F. If 10 Statistics students are selected at random, what is the probability that at least 8 of them are registered voters?

Paper For Above instruction

The purpose of this analysis is to utilize data sets related to gender, the number of pets, and voter registration status among Statistics students to compute probabilities, construct distributions, and interpret statistical relationships. Through this comprehensive examination, we aim to develop a clear understanding of how to determine probabilities in various contexts and how they inform real-world decision-making.

Part 1: Probabilities Regarding Gender and Age Deception

Based on the provided data, the initial step involves constructing a contingency table to organize counts of students by gender and whether they lied about their age. Suppose the total number of students surveyed is 200, with 120 females and 80 males. Among these, 30 students lied about their age, with 15 females and 15 males. The table would look like this:

Gender Lied About Age Did Not Lie Total
Female 15 105 120
Male 15 65 80
Total 30 170 200

Using this data, the probability that a randomly chosen student lied about their age is:

P(lied) = total students who lied / total students = 30/200 = 3/20 = 0.15

The probability that a student lied given that they are female is:

P(lied | female) = students who lied and are female / total females = 15/120 = 1/8 ≈ 0.125

Similarly, the probability that a student lied given that they are male is:

P(lied | male) = 15/80 = 3/16 = 0.1875

To determine whether being female and lying are independent events, compare the joint probability to the product of the individual probabilities:

P(female) = 120/200 = 3/5 = 0.6

P(lied) = 30/200 = 3/20 = 0.15

P(female) P(lied) = 0.6 0.15 = 0.09

Since P(female and lied) = 15/200 = 3/40 = 0.075, which is not equal to 0.09, the events are dependent. Therefore, being female influences the likelihood of lying about one's age.

Part 2: Discrete Probability Distribution for Number of Pets

Suppose we have data from 50 households indicating the number of pets: 0, 1, 2, 3, and 4 or more pets. The counts are as follows:

  • 0 pets: 10 households
  • 1 pet: 15 households
  • 2 pets: 12 households
  • 3 pets: 8 households
  • 4 or more pets: 5 households

Total: 50 households. The probability that a randomly selected household falls into each category is calculated as:

  • P(0 pets) = 10/50 = 1/5 = 0.2
  • P(1 pet) = 15/50 = 3/10 = 0.3
  • P(2 pets) = 12/50 = 6/25 = 0.24
  • P(3 pets) = 8/50 = 4/25 = 0.16
  • P(4 or more pets) = 5/50 = 1/10 = 0.1

These probabilities sum to:

0.2 + 0.3 + 0.24 + 0.16 + 0.1 = 1.0

which fulfills one condition of a probability distribution, namely, the total sum must be 1.

Calculating Mean, Variance, and Standard Deviation

Assigning numerical values for the number of pets: 0, 1, 2, 3, and 4+ (approximated as 4 for calculation purposes). The mean (expected value) is calculated as:

μ = Σ [x * P(x)] = (0)(0.2) + (1)(0.3) + (2)(0.24) + (3)(0.16) + (4)(0.1) = 0 + 0.3 + 0.48 + 0.48 + 0.4 = 1.66

Variance is computed using:

σ² = Σ [ (x - μ)² * P(x) ]

= (0 - 1.66)² 0.2 + (1 - 1.66)² 0.3 + (2 - 1.66)² 0.24 + (3 - 1.66)² 0.16 + (4 - 1.66)² * 0.1

= (2.7556)(0.2) + (0.4356)(0.3) + (0.1129)(0.24) + (1.7956)(0.16) + (5.4516)(0.1)

= 0.5511 + 0.1307 + 0.0271 + 0.2873 + 0.5452 = 1.5414

Standard deviation is the square root of the variance:

σ ≈ √1.5414 ≈ 1.241

Thus, the mean number of pets per household is approximately 1.66, with a standard deviation of about 1.24.

Probabilities of Household Pet Ownership

The probability that a household has no pets (P(not have pets)) is:

P(no pets) = 1 - P(at least one pet) = 1 - (1 - P(0 pets)) = 1 - 0.2 = 0.8

Alternatively, directly, P(no pets) = 0.2.

The probability that a household has more than 2 pets (i.e., 3 or more) is:

P(more than 2 pets) = P(3 pets) + P(4+ pets) = 0.16 + 0.1 = 0.26

Part 3: Binomial Distribution for Voter Registration

Assuming the data shows that 70% of the students are registered voters, the probability that a student is a registered voter is:

P(registered) = 0.7

Constructing the binomial distribution for the number of registered voters among 10 students involves calculating probabilities for x = 0 to 10 using the binomial formula:

P(X = x) = C(10, x)  (0.7)^x  (0.3)^{10 - x}

The mean (expected number of registered voters) is:

μ = n p = 10 0.7 = 7

Variance is:

σ² = n p (1 - p) = 10 0.7 0.3 = 2.1

Standard deviation:

σ = √2.1 ≈ 1.45

The probability that all 10 students are registered voters:

P(all registered) = (0.7)^10 ≈ 0.0282

The probability that fewer than 5 are registered:

P(X (0.7)^x (0.3)^{10 - x} ≈ 0.17

The probability that at least 8 are registered:

P(X ≥ 8) = P(8) + P(9) + P(10) ≈ 0.233 + 0.1216 + 0.0282 ≈ 0.3828

Conclusion

This analysis illustrates how probabilities can be derived from data, distributions constructed, and statistical measures computed to interpret information about student behaviors and household characteristics. The calculated probabilities, means, variances, and standard deviations provide valuable insights into patterns and dependencies within the data, facilitating informed decision-making and fostering statistical literacy.

References

  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis (3rd ed.). Cengage Learning.
  • Moore, D. S., & McCabe, G. P. (2012). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman.
  • Ott, L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • Schervish, M. J. (2012). Theory of Statistics. Springer.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage.
  • Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis (6th ed.). Pearson.

Related Assignments