Hypothesis Testing and Power Analysis

This assignment focuses on applying hypothesis testing and understanding the concept of statistical power. It involves analyzing data from a production process to determine if the mean weight of containers is significantly different from a hypothesized value of 50 ounces. The analysis utilizes a t-test and calculates the p-value, while also exploring the impact of sample size on the power of the test.

STATISTICS FOR BUSINESS

Contents
INTRODUCTION.....................................................................................................................................3
Q.1 Calculation of Covariance and correlation........................................................................................3
Q.2. Analysis and calculation of value of
.............................................................................................3
Q.3 Analysis of probability by random sampling method.......................................................................4
Q.4 Hypothesis test..................................................................................................................................4
CONCLUSION..........................................................................................................................................5
REFERENCES..........................................................................................................................................6

INTRODUCTION
Statistical analysis is considered as an evaluation of records, information and the numeric
studies in graphical and tabular form. Business statistical form help determining the stability of
business for getting effective results (Silverman, 2018). There are four questions illustrated to
clarify the concept of covariance, coefficient correlation by two different observations.
Probability of events occurs for specified duration and subject are evaluated in second question.
Hypothetical analysis with some approximate results is evaluated to an analysis the result and
relationship between the figures.
Q.1 Calculation of Covariance and correlation
Covariance: Covariance is a measurement of the combined inconsistency of two random
variables. This is a measurement tool helps in analyzing the relation between two different
variable remains linearly transformed. In statistical form the product of the deviation of two
variables form their respective means are considered as covariance (Razek, Fathy and Gawad,
2011).
Coefficient of correlation: It indicates the statistical result regarding the relationship of
independent variable to dependent variable (Huitema, 2011).
Summary of question
A sample of eight observations are given as follows
X (Experience in years) Y (Salaries in thousand)
5 20
3 23
7 15
9 11
2 27
4 21
6 17
8 14
Covariance -11.125
Coefficient of
correlation
-0.99109
a) As per above analysis the calculation Covariance for the observations are calculated as -11.125
b) Main reason of getting negative covariances is irrelevancy of both the variables. As per above
evaluation it is seen that the X variable shows the figures related to years the Y variable shows the
salaries in thousands. Both the variable corresponds shows consistency related to figures.
c) Coefficient of correlation subject to above observations are considered as -0.99109
d) One variable comparatively is greater than the other variable and the main reason of getting the
negative coefficient of correlation is greater value of variable Y than the value of X.
Q.2. Analysis and calculation of value of

Summary

 Random variable and best approximated exponential mean are equal to 3 minutes
a) If the value of
, the parameter of the exponential distribution in this situation
Let T = passed time before the next customer arrives. The random variable of T follows an
exponential distribution where
= 3 with the mean = 1/3 between the customers. The probability
density function of T is f(t) = 3e-3t where t = or > 0 minutes.
b) If customer hold more than 1.5 minutes will hang up before placing an order, 0.9898 proportion P
(X> 1.5) = e-0.5= 0.6065
c) Waiting time at which only 10%
P(X> x) = e-λ x ⇒ e-x/3 = .10 ⇒x= 6.908 minutes
d) Hold for 3 to 6 minutes
P (3 < X< 6) = e-3/3-e-6/3= e-1-e-2= 0.2325
Q.3 Analysis of probability by random sampling method
Summary
 H0: 950 hours; H1: 950 hours
 A random sample of 25 Rechargeable batteries
 Normal population with standard deviation of 200 hours
a) Calculation of
, the probability of a type II error when
 = 1000 and
 = 0.10
β= P (884.2 < < 1015.8 given that μ= 1000) = P (-2.9 < z < .40) = .6535
the results are based upon hypothesis testing and the results shows the .6535
b) Power of the test when
 = 1000 and
= 0.10
Power = 1 -β = 1 -0.6535 = 0.3465
c) Interpretation of power of test
The probability of detecting that the mean lifetime is not 950 hours, when indeed the
lifetime is 1,000 hours, is 0.3465, when α= 0.10.
d) Assessment of the results and analyzation of effect of increasing the sample size on the value of

?
Β decreases as n increases
Q.4 Hypothesis test
Summary
 Historical standard deviations of 6 ounces are given subject to production filling operation
 Mean weight for the production process is 47 ounces
 Random 36 containers for analyzing the sample mean to see the process in proper adjustment
 Standardized test if the sample mean filling weight is 48.6 ounces
 The hypothesis at the 5% level
Solution

H0: 50 ounces
H1: u is not equal to 50 ounces
x-bar = 48.6 ounces
t (48.6) = (48.6 - 50) / [6/sqrt (36)]
= -1.4
P value = 2 * P (t < -1.4 when df = 35) = 0.1703
Since the value of P is greater than 5% does not met the condition of H0. The mean of the
production process statistically similar to 50; it does not require adjustments.
CONCLUSION
The above analysis summaries the concept of business statistics. Covariance and
coefficient of correlation clarify the dependency and independency of two different variables,
whereas the probability shows the significant difference between the variables and figures on the
basis of hypothetical analysis.

REFERENCES
Books and Journals:
Silverman, B. W., 2018. Density estimation for statistics and data analysis. Routledge.
Siegel, R., Ma, J., Zou, Z. and Jemal, A., 2014. Cancer statistics, 2014. CA: a cancer journal for
clinicians. 64(1). pp.9-29.
Siegel, R., DeSantis, C. and Jemal, A., 2014. Colorectal cancer statistics, 2014. CA: a cancer
journal for clinicians, 64(2). pp.104-117.
Huitema, B., 2011. The analysis of covariance and alternatives: Statistical methods for
experiments, quasi-experiments, and single-case studies (Vol. 608). John Wiley & Sons.
Razek, A. A. K. A., Fathy, A. and Gawad, T. A., 2011. Correlation of apparent diffusion
coefficient value with prognostic parameters of lung cancer. Journal of computer assisted
tomography. 35(2). pp.248-252.