Demonstrate Inferences using a Single Sample Economics Math Problem

Demonstrate Inferences Using a Single Sample
Student’s Name
Institutional Affiliation
Demonstrate Inferences Using a Single Sample
Assignment 4
1 What is the confidence level for each of the following confidence intervals for µ?
From the z-tables, read for the z value below and multiply the table value by 2. So these are the answers:-
1 x±1.96δn= 95%
2 x±1.645δn= 90.1% when z is 1.65
3 x±2.575δn= 99.02% when z is 2.58
4 x±1.282δn= 79.94% when z is 1.28
5 x±0.99δn= 67.78%
2 A random sample of n measurements was selected from a population with unknown mean µ and standard deviation Ϭ. Calculate a 95% confidence interval for µ for each of the following situations:
The population mean µ and standard deviation or error Ϭ are both unknown. All sample sizes (n) are large. So we use a normal or Z-distribution. The formula is: x±z*δn
6 n= 75, x = 28, s2 = 12…………C.I= (27.22,28.784)
7 n= 200, x = 102, s2 = 22………C.I= (101.3499,102.6501)
8 n= 100, x = 15, s = 3………….C.I=(14.412,15.588)
9 n= 100, x =4.05, s = 0.83……...C.I= (3.88732, 4.21268)
10 Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a -d? Explain.
The assumption is true. All confidence intervals are about the population mean (x). The deviation is centered about the mean and the spread is determined by the standard deviation (Hoerl & Snee, 2012).
3 Named for the section of the 1978 Internal Revenue Code that authorized them, 401(k) plans permit employees to shift part of their before-tax salaries into investments such as mutual funds. One company, concerned with what it believed was a low employee participation rate in its 401(k) plan, sampled 30 other companies with similar plans and asked for their 401(k) participation rates. The following rates (in percentages) were obtained:
807681778280856080798270
888580798375877880847275
908482777586
Descriptive statistics from SPSS for the data are as follows:
Number of Valid Observations (Listwise) = 30
Variable Mean Std Dev Minimum Maximum N Label
PARTRATE 79.73 5.96 60.00 90.00 30
* Use the SPSS information above to construct a 95% confidence interval for the mean participation rate for all companies that have 401(k) plans.
Answer is C.I = (77.5972, 81.8628)
* Interpret the interval in the context of this problem.
The confidence interval in this problem contains information about the data parameters. It indicates that 28.5 or 29 out of 30 produced confidence intervals that contain the participation parameter. It bounds the mean and the standard deviation and if this was repeated over and over, 95% of the time participation rates would fall in between 77.5972 and 81.8628.
* What assumption is necessary to ensure the validity of this confidence interval?
The assumption is that sample size (n=30) is sufficient, observations are normally distributed, i.i.d (identicall...