Reference no: EM133189701

MTH 230 Probability and Statistics Assignment -

Question 1 - Estimation and Hypothesis Tests-One Population

Part I - An IQ test was administered to twelve students and the scores were as follows:

Assume that IQ is normally distributed with mean μ and variance σ².

1. If σ² = 196, construct a 95% confidence interval for μ, the true mean IQ in the population.

2. What sample size is necessary to ensure an interval width w = 10?

3. Now consider σ² unknown. Is the claim that μ is less than 100 justified? Perform a test at the α = 0.10 level of significance, by the following 5 steps:

Step 1. Hypotheses and significance level:

H₀:      H_a:      α =

Step 2. Test Statistic and its sampling distribution under H₀:

Step 3. Observed value of the test statistic:

Step 4. P-value under H₀:

Step 5. Decision and conclusion (interpretation of the result):

Part II - In an initial drugs test, 5.8% of job applicants tested negative. It is claimed that now the rate is lower. A random sample of 1520 current job applicants ended in 58 negative results.

1. Construct a 95% confidence interval for p, the true proportion of negative results in the population (classical form).

2. Conduct a statistical test of hypothesis on this claim at the 0.01 level of significance, by the same 5 steps as above.

Question 2 - Estimation and Hypothesis Tests-Two Populations

Part I - In a survey, two independent random samples of workers of sizes m = 124 and n = 130 from two construction firms, were asked if they wear security helmets all the time. In the first sample 106 said yes, while in the second 98 said yes. Let p₁ and p₂ be the population proportions of workers of the two firms who wear their security helmets all the time.

1. Test the equality of p₁ and p₂ at α = 0.05 (large sample procedure), by the same 5 steps as before.

2. Construct a large sample 90% confidence interval for p₁ - p₂.

Part II - The following data is on the completion time of a machining task taken by a random sample of 12 trainees and a random sample of 15 professional machinists (independent samples):

Let μ₁ and μ₂ be the corresponding population mean completion times, and assume that the populations from which these completion times were sampled are normally distributed.

1. Are the professional machinists faster on the average than the trainees (μ₁ > μ₂) at the 1% level of significance? Test this by the same 5 steps as before.

2. Construct a 99% confidence interval for μ₁ - μ₂.

Question 3 - Simple Linear Regression Analysis

The following table shows data on X = pressure of extracted gas (microns) and Y = extraction time (min). Assume Y is normally distributed with constant variance σ² for each x.

1. Construct a scatter plot of the data. Does it support the use of the simple linear regression model? Explain.

2. Calculate β^{^}₁ and β^{^}₀ (the point estimates of the slope and intercept of the regression line), and write down the estimated regression line equation of Y on X.

3. Calculate a point estimate of the standard deviation σ of Y, then calculate and interpret the coefficient of determination r².

4. Do a test of linear regression model utility at α = 0.01 (same 5 steps).

5. Calculate a point estimate of the true average extraction time when the pressure of extracted gas is x^{^} = 500 microns, then calculate a 95% PI for a future value of extraction time when the pressure of extracted gas is x^{^} = 500 microns.

6. Calculate and interpret the sample correlation coefficient r, then test (same 5 steps) for the absence of correlation between extraction time and pressure of extracted gas at α = 0.05.