Stat 350 Spring 2017 Homework 4 20 Points 1practice Problems ✓ Solved

Stat 350 Spring 2017 Homework 4 20 Points 1practice Problems

In Illinois, a typical DUI (Driving Under the Influence) offender is a 34-year-old male, arrested between 11 p.m. and 4 a.m. on a weekend, and has a BAC (blood alcohol content) of 0.16. Eighty-five percent of all drivers arrested in Illinois for driving under the influence are first-time offenders (it is their first time being arrested for DUI). Suppose 40 people arrested for DUI in Illinois are selected at random.

a) What is the probability that exactly six people are repeat offenders (they are NOT first-time offenders)?

b) What is the probability that at least two people are repeat offenders? (include at least 3 decimal places in your answer)

c) What is the mean of the number of repeat offenders that are arrested?

d) What is the standard deviation of the number of repeat offenders that are arrested?

Additional Problems: 5.87, 5.93, 5.95 Practice Problems: 5.109

Buchtal, a manufacturer of ceramic tiles, reports on average 3.9 job-related accidents per year. Accident categories include trip, fall, struck by equipment, transportation, and handling. The number of accidents is approximately Poisson.

a) What is the probability that there are exactly 3 accidents in a year?

b) Suppose a half-year is randomly selected. What is the expected number of accidents in this time period?

c) What is the standard deviation of the number of accidents in half a year?

d) If the number of accidents is more than 2 in a half-year, the company insurance carrier will raise the rates. What is the probability of an increase in the company’s insurance bill?

Additional Problems: 5.119, 5.121 Practice Problems: 6.1, 6.3

3. Determine if each of the following functions are legitimate density curves. Please graph or sketch each one.

a) f(x) = − ð‘¥2) for 0 < x < pts.

b) f(x)= 8 ð‘¥3 for x> pts.

4. The following function is a density function where k is a constant: f(x) = k(x2 + 3) for -1 < x < 3. What is the value of k?

5. The following function is a legitimate density function: ð‘“(ð‘¥) = ð‘¥ for 2 < x < 4 and 0 else.

a) Find P(1 ≤ x < 3).

b) What is the expected value?

c) Determine the cdf.

d) Find the 75th percentile.

e) Find the variance.

f) Find the standard deviation.

Practice Problems: 6.27, 6.29

6. Movie trailers are designed to entice audiences by showing scenes from coming attractions. Several trailers are usually shown in a theater before the start of the main feature, and most are available via the Internet. The duration of a movie trailer is approximately normal, with mean 150 seconds and standard deviation 30 seconds.

a) Find the probability that a randomly selected trailer lasts between 1 minute 59 seconds and 3 minutes 13 seconds.

b) Any movie trailer that lasts beyond 4 minutes and 10 seconds is considered too long. What proportion of movie trailers is too long?

c) What length will be in the top 11% of all movie trailers?

d) Find a symmetric interval about the mean such that 97% of all movie trailer durations lie in this interval.

Additional Problems: 6.43, 6.45, 6.47, 6.49 Practice Problems (from the online chapter 6.5): 6.127, 6.129

7. There have been many science conferences and research papers concerning global warming. Several climate model predictions include higher surface temperatures, rising sea levels, and larger subtropical deserts. Global leaders continue to discuss possible actions to stop or slow these trends. Despite the ominous warnings, a recent survey indicated that only 44% of Americans said that global warming should be a high priority for political leaders and governments. Suppose 250 Americans are selected at random and each is asked if global warming should be a high priority issue. Answer each problem using the normal approximation to the binomial distribution.

a) Is the normal approximate to the binomial appropriate in this situation? Do part b) no matter what your answer is.

b) Find the approximate probability that at most 110 Americans believe global warming should be a high priority issue. Remember the continuity correction.

Paper For Above Instructions

In the analysis of the DUI offenders in Illinois, we need to find the probability of exactly six repeat offenders among a sample of 40 individuals where first-time offenders compose 85% of the group (the remaining 15% being repeat offenders). Thus, this scenario aligns with a binomial distribution since it involves a fixed number of trials, two possible outcomes (repeat offenders vs. first-time offenders), and a constant probability of success.

Let X represent the number of repeat offenders. Here, X ~ Binomial(n=40, p=0.15). The probability of finding exactly six repeat offenders can be calculated using the binomial probability formula:

P(X=k) = C(n,k) (p^k) (1-p)^(n-k)

Substituting for k = 6:

P(X=6) = C(40,6) (0.15)^6 (0.85)^(34)

Where C(40,6) is the binomial coefficient. Let's calculate this:

C(40, 6) = 40! / (6!(40-6)!) = 40! / (6! * 34!) = 3,838,380.

The probability calculation yields:

P(X=6) = 3,838,380 (0.15)^6 (0.85)^(34) ≈ 0.227.

Next, for part (b), we need to determine the probability that at least two are repeat offenders:

P(X ≥ 2) = 1 - P(X=0) - P(X=1).

Calculating P(X=0) and P(X=1):

P(X=0) = C(40,0) (0.15)^0 (0.85)^40 ≈ 0.027.

P(X=1) = C(40,1) (0.15)^1 (0.85)^(39) ≈ 0.153.

Thus, P(X ≥ 2) = 1 - (0.027 + 0.153) ≈ 0.820.

For the mean and standard deviation for the number of repeat offenders:

Mean, μ = n p = 40 0.15 = 6.

Standard deviation, σ = √(n p (1-p)) = √(40 0.15 0.85) ≈ 3.46.

In the case of Buchtal's accidents, the yearly average is characterized by a Poisson distribution, with λ = 3.9 accidents per year. For part (a), to find the probability of exactly three accidents:

P(X=k) = (e^(-λ) * (λ^k)) / k!.

P(X=3) = (e^(-3.9) * (3.9^3)) / 3! ≈ 0.241.

For part (b), when considering a half-year, we would expect λ to be halved:

Expected accidents in half a year: λ = 3.9 / 2 = 1.95.

For part (c), the standard deviation in half a year is simply the square root of λ: σ = √1.95 ≈ 1.40.

For part (d), the probability of more than two accidents in half a year can be calculated as:

P(X > 2) = 1 - (P(X=0) + P(X=1) + P(X=2)),

where we can compute each of these using the above Poisson formula.

Moving to the functions in density curves, we'll focus on ensuring that total area equates to 1 for valid functions. Applications of integration and ensuring non-negativity will be crucial in validating density functions as legitimate.

Continuing to the movie trailers, we can utilize z-scores to process the duration distribution. For instance, to establish the probability that a trailer lasts between 119 and 193 seconds:

Calculating z-scores:

z = (X - μ) / σ, resulting in z-scores for each time boundary, subsequently referencing z-tables for a probability result.

Lastly, the global warming question applies normal approximation to binomial distribution retrieving relevant probabilities based on the selected sample size of 250 respondents, with completion of calculations showcasing how public opinion gauges on prioritizing climate change.

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