Statistics and Probability Answers

From a random sample of 22 cities of various sizes, data is collected from police records regarding violations of the narcotic drug laws per 100,000 inhabitants. The communities were broken down into 4 sizes (large cities, small cities, suburban communities, and rural communities), and data was collected over the period of one calendar year. Please determine if there is significant evidence to conclude that the mean rates of arrests for narcotics is different in each community (based on their size). Please conduct this test at the 99% confidence level.

City Community
Large: 45, 34, 41, 42, 37 Suburban: 25, 17, 19, 28, 31, 37
Small: 23, 24, 27, 21, 26, 34 Rural: 13, 16, 14, 17, 10

 Determine the direction of the hypothesis test (one-sided left, one-sided right, bidirectional)
 Determine the test statistic (z* or t*) and the p-value for each of the following situations and
 Determine if they would cause the rejection of the null hypothesis if the confidence level was set
at 95% in each case. (Hint: be wary of the sample size)

Ho: μ = 380 s, Ha: μ ≠ 380 s, sample mean = 357, s = 75, n = 40

A manufacturer of coffee vending machines has designed a new, less expensive machine. The current machine is known to dispense an average of 6 fl. oz., with a standard deviation of .2 fl. oz., into cups. When the new machine is tested using 15 cups, the mean and the standard deviation of the fills are found to be 6 fl. oz. and .214 fl. oz. Test H0: σ = 0.2 versus Ha: σ ≠ 0.2 at levels of significance .05 and .01. Assume normality. (Round your answer to 4 decimal places.)

chi-square

You have been asked to compare two neighbourhoods in the west end of Montreal by a real estate agency. The agency wants to know if a difference exists in the selling price of houses on Westminster street (Montreal West), as compared to Beaconsfield street (NDG). You take a sample of houses sold on both streets over the past 10 years, adjust the inflation and monetary rate so as to equalize the dollar, and tabulate your results. Is there enough evidence to prove that the selling price for the houses on the two streets is significantly different? Please conduct a complete hypothesis test at the 99% confidence level and calculate the p-value. (Hint: Can we assume that the variances are equal?)

Street n x s
Westminister 16 125,950 2400
Beaconsfield 24 128,800 3700

A group of ten students from the course INTE 296 was approached by a famous scientist by the name of Dr. Wild E. Coyote with an experimental drug to aid in their retention of statistical procedures. The drug, called MemoraidTM, was administered after the students completed a generic, comprehensive exam. Twenty-four hours later, the students were given a similar exam, and the test scores were compared. Based on the following table, is there enough evidence to conclude that the drug was effective in increasing their statistical ability? Use an alpha value of 0.05 and find the p-value.

Student: 1 2 3 4 5 6 7 8 9 10
Before: 75 62 70 70 55 59 60 64 72 65
After: 74 65 68 74 62 62 60 67 75 58

Please find the limit(s) of the boundaries for the critical region for the following hypotheses (You only need to find the z, t, or F-critical value in all cases, e.g., +/2.33. Please do not complete the hypothesis test):
a. Ho: μd ≤ 6.5, Ha: μd > 6.5, α = 0.05, n = 35, s = 2.5.
b. Ho: μ1 - μ2 ≤ 0, Ha: μ1 - μ2 > 0, α = 0.1, n1 = 13, n2 = 17 (σ1 = σ2).
c. Ho: μ1 - μ2 ≥ 0, Ha: μ1 - μ2 < 0, α = 0.025, n1 = 19, n2 = 27 (σ1 ≠ σ2).
d. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.01, n1 = 31, n2 = 21 s1 = 4.1, s2 = 8.3.
e. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.05, n1 = 22, n2 = 13 s1 = 7.2, s2 = 10.6.

Suppose we conduct a poll to estimate the proportion of voters who favor a major presidential candidate. Assuming that 50 percent of the electorate could be in favor of the candidate, determine the sample size needed so that we are 95 percent confident that formula117.mml, the sample proportion of voters who favor the candidate, is within a margin of error of .01 of p, the population proportion of all voters who are in favor of the candidate.

a) The distribution of marks obtained by 500 candidates in a particular exam is given
below:

Marks more than: 0 10 20 30 40 50
Number of candidates 500 460 400 200 100 30

Calculate the lower quartile marks. If 70% of the candidates pass in the exam, find
the minimum marks obtained by a pass candidate. (5)
b) An analysis of monthly wages paid to the workers of two firms A and B belonging
to the same industry gives the following results:

Firm A Firm B
Number of workers 500 600
Average daily wages ` 186 ` 175
Variance of distribution of wages 81 100

i) Which firm, A or B, has a large wage bill?
ii) In which firm, A or B, is there greater variability in individual wages?
iii) Find the average daily wage and the variance of the distribution of wages of all
the workers in the firms A and B taken together.

a) Let the random variable X have the following distribution:
P( and X = )0 = P(X = )2 = p
P(X = )1 =1− 2 p where
2
1
0 ≤ p ≤
for what value of p is the ) var(X a maximum? Justify. (3)
b) In a binomial distribution consisting of 5 independent trials, probabilities of 1 and 2
successes are 0.4096 and 0.2048 respectively. Find the parameter ‘ p ’ of the
distribution. (3)
c) Fit a Poisson distribution to the following data: (4)

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a) Let X be normally distributed and the mean of X is 12 and S.D. is 4. Find
i) ] P[X ≥ 20
ii) ] P[X ≤ 20
iii) ] p 0[ ≤ X ≤12
iv) x′ when 24.0 p[X > x′] =
v) 0
x′ and 1
x′, when 50.0 ( ) P x0
′ < X < x1
′ = and 25.0 ( ) P X > x1
′ = . (6)
b) A die is thrown 9000 times and the outcome of 3 or 4 is observed 3240 times. Show
that the die cannot be regarded as an unbiased one and find the limits between which
the probability of a throw of 3 or 4 lies