SOLUTION: A nationwide job recruiting firm wants to compare the annual incomes for childcare workers in Arizona and Nevada. Due to recent trends in the childcare industry, the firm suspects
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Question 1192251: A nationwide job recruiting firm wants to compare the annual incomes for childcare workers in Arizona and Nevada. Due to recent trends in the childcare industry, the firm suspects that the mean annual income of childcare workers in Arizona is less than the mean annual income of childcare workers in Nevada. To see if this is true, the firm selected a random sample of 10 childcare workers from Arizona and an independent random sample of 10 childcare workers from Nevada and asked them to report their mean annual income. The data obtained were as follows. Annual income in dollars Arizona 45325, 48581, 41694, 34501, 32665, 28179, 38930, 45174, 31800, 43875 Nevada 43648, 42805, 36433, 37896, 35588, 50942, 47219, 40051, 50079, 42779 Send data to calculator Send data to Excel The population standard deviation for the annual incomes of childcare workers in Arizona and in Nevada are estimated as 6300 and 6500, respectively. It is also known that both populations are approximately normally distributed. At the 0.05 level of significance, is there sufficient evidence to support the claim that the mean annual income, u1, of childcare workers in Arizona is less than the mean annual income, u2, of childcare workers in Nevada? Perform a one-tailed test. Then complete the parts below.
Carry your intermediate computations to at least three decimal places. (If necessary, consult a list of formulas.)
(a) State the null hypothesis and the alternative hypothesis .
(b) Determine the type of test statistic to use.
(c) Find the value of the test statistic. (Round to three or more decimal places.)
(d) Find the critical value at the level of significance. (Round to three or more decimal places.)
(e) Can we support the claim that the mean annual income of childcare workers in Arizona is less than the mean annual income of childcare workers in Nevada?
Yes or No
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
(a) State the null hypothesis H0 and the alternative hypothesis H1.
H0: μ1 = μ2 (The mean annual income of childcare workers in Arizona is equal to the mean annual income of childcare workers in Nevada)
H1: μ1 < μ2 (The mean annual income of childcare workers in Arizona is less than the mean annual income of childcare workers in Nevada)
(b) Determine the type of test statistic to use.
Since the population standard deviations are known and the populations are approximately normally distributed, we will use a z-test for the difference between two population means.
(c) Find the value of the test statistic. (Round to three or more decimal places.)
To calculate the z-test statistic, we can use the following formula:
```
z = (x̄1 - x̄2) / √((σ1²/n1) + (σ2²/n2))
```
Where:
* x̄1 and x̄2 are the sample means for Arizona and Nevada, respectively
* σ1 and σ2 are the population standard deviations for Arizona and Nevada, respectively
* n1 and n2 are the sample sizes for Arizona and Nevada, respectively
Plugging in the given values:
```
z = (39495.7 - 43425.9) / √((6300²/10) + (6500²/10))
z = -3930.2 / √((39690000/10) + (42250000/10))
z = -3930.2 / √(8194000)
z ≈ -1.371
```
(d) Find the critical value at the 0.05 level of significance. (Round to three or more decimal places.)
Since this is a one-tailed test with a significance level of 0.05, we need to find the z-value that corresponds to the 5th percentile in the standard normal distribution. Using a z-table or calculator, we find that the critical value is approximately -1.645.
(e) Can we support the claim that the mean annual income of childcare workers in Arizona is less than the mean annual income of childcare workers in Nevada?
No. The test statistic (-1.371) is greater than the critical value (-1.645). Therefore, we fail to reject the null hypothesis. There is not sufficient evidence to support the claim that the mean annual income of childcare workers in Arizona is less than the mean annual income of childcare workers in Nevada.
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