Assignment: Solve The Following Statistics Problems 1. The ✓ Solved

Assignment: Solve the following statistics problems.

1. The

Assignment: Solve the following statistics problems.

1. The price of a particular brand of jeans has a mean of $37.99 and a standard deviation of $7. A sample of 49 pairs of jeans is selected. a. What is the probability that the sample of jeans will have a mean price less than $40?

2. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The average age in the sample was 22.5 years with a standard deviation of 5 years. Construct a 98% confidence interval for the average age of all consumers.

3. A local hotel wants to estimate the proportion of its guests that are from out-of-state. Preliminary estimates are that 45% of the hotel guests are from out-of-state. How large a sample should be taken to estimate the proportion of out-of-state guests with a margin of error no larger than 6% and with a 95% level of confidence?

4. A student believes that no more than 20% of students who finish a statistics course get an A. A random sample of 100 students was taken. 24%of the students in the sample received As.

a. State the null and alternative hypotheses.

5. In the past the average age of employees of a large corporation has been 40 years. Recently, the company has been hiring older individuals. In order to determine whether there has been an increase in the average age of all the employees, a sample of 25 employees was selected. The average age in the sample was 45 years with a standard deviation of 5 years.

Let ( = .05. Test to determine whether the mean age of all employees is more than 40 years.

a. State the null and alternative hypotheses.

b. Use critical value approach. State the rejection rule. Test statistic. Conclusion.

c. Use p-value approach. State the rejection rule, p-value. Conclusion.

Paper For Above Instructions

Problem 1: Sampling distribution and probability of a sample mean

Problem 2: Confidence interval for a mean with small sample

Problem 3: Sample size for estimating a proportion

Problem 4: Hypotheses for a proportion

0.20. Observed p̂ = 0.24 with n = 100. Under H0, the test statistic is z = (p̂ − p0)/√[p0(1 − p0)/n] = (0.24 − 0.20)/√[0.20 × 0.80 / 100] = 0.04 / √0.0016 = 0.04 / 0.04 = 1.00. The one-sided critical value at α = 0.05 is z0.95 ≈ 1.645. Since 1.00 1.00) ≈ 0.1587. This illustrates how an observed proportion above 0.20 may not be statistically significant at the chosen level (Casella & Berger, 2002; Agresti & Franklin, 2017).)

Problem 5: One-sample t-test for mean age

40. With n = 25, x̄ = 45, s = 5, the t-statistic is t = (x̄ − μ0) / (s/√n) = (45 − 40) / (5/√25) = 5 / 1 = 5. Degrees of freedom are df = n − 1 = 24. The one-tailed critical value at α = 0.05 is t0.95,24 ≈ 1.711. Since 5 > 1.711, we reject H0. The p-value is P(T24 > 5), which is extremely small (p

Discussion and synthesis

References

• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Agresti, A. (2017). Statistical Methods for the Social Sciences (5th ed.). Pearson.
• Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data (3rd ed.). Pearson.
• Hogg, R. V., McKean, J., & Craig, A. (2018). Introduction to Mathematical Statistics (8th ed.). Pearson.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2013). Introduction to the Practice of Statistics (7th ed.). W.H. Freeman.
• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
• Rice, J. A. (2007). Mathematical Statistics and Data Analysis (3rd ed.). Brooks/Cole.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2013). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
• Wentworth, P. S. (various editions). Probability and Statistics for Engineers and Scientists. Pearson.
• Other foundational texts used for standard methods include: Devore, J. (2011). Probability and Statistics for Engineering and the Sciences (8th ed.). Brooks/Cole.