Problem 5

Let \(X=\left( {{X}_{1}},{{X}_{2}},...,{{X}_{{{n}_{1}}}} \right)\), \(Y=\left( {{Y}_{1}},{{Y}_{2}},...,{{Y}_{{{n}_{2}}}} \right)\) be independent random samples from the normal distributions \(N\left( {{\mu }_{X}},\sigma _{X}^{2} \right)\), \(N\left( {{\mu }_{Y}},\sigma _{Y}^{2} \right)\).

1. Normalize \(\bar{X}-\bar{Y}\)
2. Compute the realization of the normalized random variable \(\bar{X}-\bar{Y}\). You know that \(\bar{X}\) = 140 was computed from a realization with the range 60, \(\bar{Y}\) = 137 was computed from a realization with the range 50, \({{X}_{i}}\tilde{\ }N\left( 142,20 \right)\) and \({{X}_{i}}\tilde{\ }N\left( 135,18 \right)\).
   Problem 6
   Let X, Y be independent random variables \(X\tilde{\ }Bi\left( {{m}_{1}},{{\pi }_{1}} \right)\), \(Y\tilde{\ }Bi\left( {{m}_{2}},{{\pi }_{2}} \right)\).
   1. Normalize \(\frac{X}{{{m}_{1}}}-\frac{Y}{{{m}_{2}}}\)
   2. Let x = 264, y = 496. Compute the realization of the normalized random variable \(\frac{X}{{{m}_{1}}}-\frac{Y}{{{m}_{2}}}\).You know that \(X\tilde{\ }Bi\left( 300,0.85 \right)\) and \(Y\tilde{\ }Bi\left( 620,0.83 \right)\).
      Problem 7
      We know the last 10 accident benefits related to motorbike insurance:
      7, 1, 3, 5, 9, 18, 35, 1, 3, 12
      Find the empirical cumulative distribution function and plot the graph of this function
      Lecture Part
      Problem 1
      You know percentage daily increases in value of stocks in ten randomly selected days at stock market.
      10%, 16%, 5%, 10%, 12%, 8%, 4%, 6%, 5%, 4%.
      We assume that the percentage daily increase in value is a random variable with a normal distribution.
3. Test the hypothesis that the population mean of daily increases in value is greater than 7% using a significance level 0.05.
4. Test the hypothesis that the population dispersion of daily increases in value is less than 3% using a significance level 0.01.
   Problem 2
   We randomly selected 100 displays and measured their lifetime in hours (in thousands). Our results are included in the following table:

   lifetime	11	12	13	14
   number of displays	15	30	45	10

   
   Assume that the lifetime is a random variable \(X\tilde{\ }N\left( \mu ,{{\sigma }^{2}} \right)\).
   1. Find a 95% confidence interval for the population mean of display lifetime!
   2. Find the maximal population mean display lifetime with 95% confidence!
   3. Find the minimal population mean display lifetime with 90% confidence!
   4. Do the same for \({{\sigma }^{2}}\)

Problem 3

We would like to know the percentage of new mobile operator customers in Banská Bystrica. We obtained the following results by public opinion survey. There were 35 customers of the new operator from 100 randomly selected respondents.

1. Find a 95% confidence interval for the population proportion of new mobile customers!

2. Find the maximal population proportion of new mobile operator customers with 95% confidence!

3. Find the minimal population proportion of new mobile operator customers with 90% confidence!

4. Assume that the maximum tolerable error \(\Delta \) has to be < 0.01. What number of respondents do we need?

Testing hypotheses 1: One sample tests

Problem 1

A manufacturer of men’s sport shirts knows that its brand is carried in 24% of the men’s clothing stores in Bratislava. The manufacturer recently sampled 10 men’s clothing stores in Banska Bystrica and Zvolen and found that 2 stores carried their brand. At the 0.01 significance level, is there evidence that the company has poorer distribution in Bansk´a Bystrica and Zvolen?

Problem 2

A power company wants to determine whether or not mean residential electricity usage per slovak household has decreased (due to high daily temperatures) for the Winter 2007. Past experience indicates that the average usage per months is normally distributed with a mean of 700 kwh and a standard deviation of 50 kwh. A simple random sample of 50 households was selected

695, 700, 705, 699, 695, 698, 700, 690, 696, 696, 690, 692, 697, 699, 699, 698, 697, 697, 698, 699, 850, 715, 400, 550, 600, 650, 687, 730, 677, 699, 654, 650, 666, 900, 715, 830,

489, 450, 645, 590, 789, 650, 830, 745, 905, 826, 720, 560, 771, 732.

Test at the 0.05 significance level whether or not the mean residential electricity usage per household has decreased for the Winter 2007.

Problem 3

Business Month reported that women in executive positions earn just 58% as much as their male counterparts. A consulting firm reports that figure is 70%. Suppose that the consulting firm conclusion is based on a random sample of 150 women executives. Is the Business Month conclusions justified at the 0.01 significance level?

Problem 4

Under a standard manufacturing process, the breaking strength of nylon thread is normally distributed, with mean equal to 100 and a standard deviation equal to 5. A new, cheaper process is tested, and a sample of 10 threads is drawn, with the results

105, 95, 95, 99, 98, 97, 101, 100, 100, 99.

Your assistant has noted, "Since the sample mean (standard deviation) is not (is) significantly less than 100 (5) at alpha 0.05, we have strong justification for installing the new process." Do you agree? Explain!

Problem 5

A company claims that the rubber belts it manufactures have a mean service life of at least 800 hours with the standard deviation less than 5 hours. A simple random sample of 20 belts is observed:

800, 780, 790, 820, 810, 840, 750, 795, 805, 810, 820, 800, 801, 802, 803, 804, 812, 805, 801, 802. Test the assumption of the company! Assume the normally distributed service life!

Problem 6

A manager wants to ascertain whether a quality improvement program has reduced the percentage of defective units produced by a production team in the company. Before the program began, the team’s output was determined to be 15% defective. A random sample of 100 units is observed and 14 defective units are found. Test the hypothesis about effectiveness of this program! Test the same hypothesis for 140 defective units from 1000 units!

Problem 7

See Problem 7 from Problem set 1!

You know percentage daily increases in value of stocks in ten randomly selected days at stock market.

10%, 16%, 5%, 10%, 12%, 8%, 4%, 6%, 5%, 4%.

We assume that the percentage daily increase in value is a random variable with a normal distribution.

1. Test the hypothesis that the population mean of daily increases in value is greater than 7%. Assume a significance level 0.05.

2. Test the hypothesis that the population standard deviation of daily increases in value is less than 3%. Assume a significance level 0.01.

Problem 8

An issue of The Wall Street Journal reported an increase in the percentage of taxpayers who used professional preparers from 45% in 1986 to 47% in 1987. A study was conducted in 1988 to see if a significant increase in the usage of professional prepares occurred. A random sample of 1000 taxpayers was surveyed, and 494 indicated they used a professional preparer. Using the 0.1 significance level, can it be concluded that there has been an increase since 1987?

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