QuestionSuppose the weights of seventh-graders at a certain school vary according to a Normal distribution, with a mean of 100 pounds and a standard deviation of 7.5 pounds. A researcher believes the average weight has decreased since the implementation of a new breakfast and lunch program at the school. She finds, in a random sample of 35 students, an average weight of 98 pounds.
What is the -value for an appropriate hypothesis test of the researcher's claim?
0.057
0.115
0.943
Studdy Solution
STEP 1
1. The weights of seventh-graders are normally distributed with a mean () of 100 pounds and a standard deviation () of 7.5 pounds.
2. The sample size is 35 students, with a sample mean () of 98 pounds.
3. We are conducting a one-sample t-test for the mean.
4. The null hypothesis () is that the population mean is 100 pounds.
5. The alternative hypothesis () is that the population mean is less than 100 pounds.
STEP 2
1. Set up the hypotheses.
2. Calculate the test statistic.
3. Determine the -value.
4. Compare the -value to the significance level.
STEP 3
Set up the hypotheses:
- Null hypothesis ():
- Alternative hypothesis ():
STEP 4
Calculate the test statistic using the formula for the t-statistic:
where:
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-
-
-
Substitute the values:
Calculate the denominator:
Calculate the t-statistic:
STEP 5
Determine the -value for the calculated t-statistic () using a t-distribution with degrees of freedom.
The -value is approximately 0.057.
STEP 6
Compare the -value to a common significance level (e.g., ):
- If -value , reject .
- If -value , do not reject .
Since the -value is 0.057, which is slightly greater than 0.05, we do not reject .
The -value is:
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