Question

The executives at AYZ Community Behavioral Health Organization have received numerous reports over the summer about adults being brought to nearby hospitals who are found to be malnourished. One executive hypothesizes that he suspects that at least 48% of these adults must have an undiagnosed mental health condition. In response to disagreement from the rest of the executives present that the number would be closer to 25%, which is the national average for this circumstance, he commissioned a study to review the available data as a means of testing his theory. After samples of 100 inpatient records from each of 40 districts were examined, the analysis of the study revealed the following statistics:

Sample size per district = 100

Mean per district = 38%

Population Standard Deviation (for all patients admitted to all hospitals during the period who were malnourished) = 20%

Question: Was the executive who commissioned the study correct with respect to his belief that 48% of the malnourished inpatients had an undiagnosed mental health condition?

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u > 48
Alternative hypothesis: u < 48

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample z-test.

Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).

SE = s / sqrt(n)

S.E = 2.0

z = (x - u) / SE

z = - 5.0

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a z statistic test statistic of - 5.0

Thus the P-value in this analysis is less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that at least 48% of these adults must have an undiagnosed mental health condition.