Question

It is believed that the average amount of money spent per U.S. household per week on food is about $98, with standard deviation $10. A random sample of 100 households in a certain affluent community yields a mean weekly food budget of $100. We want to test the hypothesis that the mean weekly food budget for all households in this community is higher than the national average. State the null and alternative hypotheses for this test.

Answer 1

We conclude that the mean weekly food budget for all households in this community is higher than the national average of $98.

Explanation:

We are given that it is believed that the average amount of money spent per U.S. household per week on food is about $98, with standard deviation $10.

A random sample of 100 households in a certain affluent community yields a mean weekly food budget of $100.

Let
`\mu` = population mean weekly food budget for all households in this community.

SO, Null Hypothesis,
`H_0` :
`\mu`
`\leq` $98 {means that the mean weekly food budget for all households in this community is less than or equal to the national average of $98}

Alternate Hypothesis,
`H_A` :
`\mu` > $98 {means that the mean weekly food budget for all households in this community is higher than the national average of $98}

The test statistics that will be used here is One-sample z test statistics as we know about the population standard deviation;

T.S. =
`(\bar X -\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\bar X` = sample mean weekly food budget = $100

`\sigma` = population standard deviation = $10

n = sample of households = 100

So, test statistics =
`(100-98)/((10)/(√(100) ) )`

= 2

Since in the question we are not given with the level of significance to test the hypothesis, so we assume it to be 5%. Now at 5% significance level, the z table gives critical value of 1.6449 for right-tailed test. Since our test statistics is higher than the critical value of z as 2 > 1.6449, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the mean weekly food budget for all households in this community is higher than the national average of $98.