Assignment 1
Assignment 1
The mean value of the price calculates:
X=i=1nxin,
where xi is the value of the household in state i; n is the quantity of states.
The standard deviation calculates:
s=i=1n(xi-X)2n-1.
The confidence interval calculates:
CI=t(0.95;n-1)∙sn,
Where t(0.95;n-1) the t-value from the table. For the national household, t0.95;50=2.01, for the states where the company wants to open the new locations: t0.95;9=2.262.
Data for the Ten States
Median is obtained by arranging the values in ascending order and taking the central value; for the set with 10 value the median is the mean of 5th and 6th values:
112900 128400 138800 154900 158100 159300 161300 209000 235800 358800
Median = ($158,100 + $159,300) / 2 = $158,700.
The calculations for the states where the company wants to open the offices:
X=($112,900 + $128,400 + $138,800 + $154,900 + $158,100 + $159,300 + $161,300 +$209,000 + $235,800 + $358,800)/10 = $181,730
s=(112,900-181,730)2+(128,400-181,730)2++(358,800-181,730)210-1=$71,991.
CI=t(0.95;n-1)∙sn=2.262∙71,99110=$51,496.
The mean value is $181,730 ± $51,496. Therefore, with 95% confidence, we can guarantee that the average price in ten states is between $130,234 and $233,226.
Nationwide Data
The national data of the household prices include 51 samples, and therefore the calculations are reasonable to perform with MS Excel. Using the descriptive statistics the mean, median and standard deviations were obtained. Table 1 presents the results.
The Mean, Median and Standard Deviation for the National Household Prices
The confidence interval for the national data is:
CI=2.01∙85858.751=24,165.47.
Nationwide, the mean household price changes as Xnat±CI. This is $193,190.2 ± $24,165.47. Therefore, with 95% confidence, we can guarantee that the average nationwide price is between $169,024.7 and $217,355.7.
Hypothesis Testing
The null hypothesis H0: the mean values of the nationwide households and in the states where the company is going to open offices are equal, Xnat=X.
The alternate hypothesis H1: the mean value of the nationwide households is less than in the states where the company is going to open offices, Xnat<X.
The decision is made basing on one-sided t-test. Therefore, the one-sided t-value is used as a critical value. The number of degrees of freedon is f = n + nnat – 2 = 10 + 50 – 2 = 58, and t-value is t0.95;58=1.68.
t=Xnat-Xsc;
sc=snat2+s2.
Calculation:
sc=85,858.72+71,9912=$112,046.52
t=193,190.2-181,730112,046.52=0.10;
Comparing t(0.95;58) and t, t(0.95;58) < t (1.68 < 0.1). Therefore, the null hypothesis is true and the mean values of the nationwide households and in the states where the company is going to open offices are equal.
Analysis if States Prices are Within the Confidence Interval
The results are listed in the table 2.
Household Prices and the Population Confidence Interval
Therefore, the salaries are expected to rise only in two states: Colorado and California. The re-location for the majority of the states will not be associated with salary increase.
