A researcher conjectures that cities in the more populous states
of the United States tend to
have higher costs for hospital rooms. Using “city data” that
accompany this text, select a
random sample of ten cities from the six most populous states
(California, Texas, New York,
Florida, Pennsylvania and Illinois). Then take a random sample of
ten cities from the remaining
states in the data set. For each of the twenty cities, record the
average daily cost of a private
hospital room. Assume that such costs are approximately normally
distributed for all cities in
each of the two groups of states. Further assume that the cities
you selected make random
samples of all cities for the two groups of states and that the
standard deviations for the two
groups are unequal and unknown. (When samples are unequal and
unknown, it is said that they are
independent of each other. So when calculating the degrees of
freedom use the formula below
to calculate df. You still use the same confidence interval and t -
test statistic as per your
notes).
Steps for Hypothesis Test for Independence: (σ1 and σ'2unknown
and unequal)
(The t - distribution)
1. state the null and alternative hypothesis
2. find the degrees of freedom when the population standard
deviations are unknown and
unequal.
3. find your critical value(s) of t given the level of significance
and degrees of freedom.
4. find the estimate of the standard deviation.
5. find the test statistic for t.
6. Find the confidence interval using 95% for the “city data”
7. make a decision with a brief summary of your data analysis
HELP ME PLEASE!!!
To carry out the hypothesis test assuming unequal variance for the cities data, the Welch's two sample t test need to be performed to compare the two sample means while assuming unequal variance.
(First take random sample of cities data, let
Group 1: random sample of ten cities from the six most populous states (California, Texas, New York, Florida, Pennsylvania and Illinois) where mean = and standard deviation = and sample size, = 10
Group 2: random sample of ten cities from the other states where mean = and standard deviation = and sample size, = 10
Now, the test is performed in following steps,
Step 1: The Null and Alternative Hypotheses
Step 2: Select the appropriate test statistic and level of significance.
The t statistic is used to compare the two population means. The t statistic is computed using the formula,
This t-statistic follows a t distribution with degrees of freedom
Step 3: State the decision rules.
The decision rules state the conditions that if,
,
Step 4: Compute the appropriate test statistic and make the decision.
The t statistic is,
The P-value for the t statistic is obtained using the t distribution table for t and degree of freedom = df
Decision:
If the null hypothesis rejected, we can support the claim that cities in the more populous states of the United States tend to have higher costs for hospital rooms
The confidence interval for difference in means is obtained using the formula,
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