Question

The US Agency for Healthcare Research and Quality (www.ahrq.gov) reported that in 2010 the mean cost of a stay in a hospital for American women aged 18-44 was $15,200. A random sample of 400 hospital stays for American women aged 18-44 showed a mean cost of $16,000, with a standard deviation of $5000. Test whether the population mean cost for a hospital stay for American women aged 18-44 has increased since 2010, using the 5% level of significance.

Answer 1

We conclude that the population mean cost for a hospital stay for American women aged 18-44 has increased since 2010.

Explanation:

We are given that the US Agency for Healthcare Research and Quality reported that in 2010 the mean cost of a stay in a hospital for American women aged 18-44 was $15,200.

A random sample of 400 hospital stays for American women aged 18-44 showed a mean cost of $16,000, with a standard deviation of $5000.

Let
`\mu` = population mean cost for a hospital stay for American women aged 18-44.

SO, Null Hypothesis,
`H_0` :
`\mu`
`\leq` $15,200 {means that the population mean cost for a hospital stay for American women aged 18-44 has reduced or remains same since 2010}

Alternate Hypothesis,
`H_A` :
`\mu` > $15,200 {means that the population mean cost for a hospital stay for American women aged 18-44 has increased since 2010}

The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;

T.S. =
`(\bar X -\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean cost of a stay in a hospital for American women aged 18-44 = $16,000

s = sample standard deviation = $5,000

n = sample of hospital stays = 400

So, test statistics =
`(16,000-15,200)/((5,000)/(√(400) ) )` ~
`t_3_9_9`

= 3.20

Now at 5% significance level, the t table gives critical value of 1.645 at 399 degree of freedom for right-tailed test. Since our test statistics is more than the critical value of t as 1.645 < 3.20, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the population mean cost for a hospital stay for American women aged 18-44 has increased since 2010.