Question

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!!!

Answer 1

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 = $\overline{X}_1$ and standard deviation = $s_1$ and sample size, $N_1$ = 10

Group 2: random sample of ten cities from the other states where mean = $\overline{X}_2$ and standard deviation = $s_2$ and sample size, $N_2$ = 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,

2

Where, X, mean of sample. s2. variance of sample and . sample size

This t-statistic follows a t distribution with degrees of freedom

df =

Step 3: State the decision rules.

The decision rules state the conditions that if,

,

P-value < α 0.05 Reject the null hypothesis

P-value > α-0.05 Failed to reject the null hypothesis

Step 4: Compute the appropriate test statistic and make the decision.

The t statistic is,

2

df =

The P-value for the t statistic is obtained using the t distribution table for t and degree of freedom = df

Decision:

Reject the null hypothesis if P-value < α .05

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,

/ + Confidence Interval = (X1-X2) ± t*