A manufacturer of chocolate candies uses machines to package candies as they move along a filling line. Although the packages are labeled as 8​ ounces, the company wants the packages to contain a mean of 

8.178.17 

ounces so that virtually none of the packages contain less than 8 ounces. A sample of 

5050 

packages is selected​ periodically, and the packaging process is stopped if there is evidence that the mean amount packaged is different from 

8.178.17 

ounces. Suppose that in a particular sample of 

5050 

​packages, the mean amount dispensed is 

8.1658.165 

​ounces, with a sample standard deviation of 

0.0460.046 

ounce. Complete parts​ (a)**** and​ (b).  Is there evidence that the population mean amount is different from 

8.178.17 

​ounces? (Use a 

0.050.05 

level of​ significance.)

State the null and alternative hypotheses.

Upper H 0H0​: 

muμ

▼
 
not equals≠

equals=

less than<

less than or equals≤

greater than or equals≥

greater than>
 
enter your response here

Upper H 1H1​: 

muμ

▼
 
less than or equals≤

equals=

less than<

not equals≠

greater than or equals≥

greater than>
 
enter your response here

### (a) **State the null and alternative hypotheses:**

We are testing whether the mean amount of chocolate candies in the packages is different from the target mean of \( 8.17 \) ounces. Therefore, we are performing a two-tailed hypothesis test.

- **Null hypothesis (\( H_0 \))**: The population mean amount of candy is equal to 8.17 ounces.
  \[
  H_0: \mu = 8.17
  \]
  
- **Alternative hypothesis (\( H_1 \))**: The population mean amount of candy is not equal to 8.17 ounces (indicating a two-tailed test).
  \[
  H_1: \mu \neq 8.17
  \]

---

Would you like further details on the next steps, such as calculating the test statistic or making a conclusion based on the \( p \)-value or critical value? 

Here are 5 relative questions to expand your understanding:

1. How do we calculate the test statistic for a two-tailed hypothesis test?
2. What is the decision rule based on the \( p \)-value and significance level \( \alpha = 0.05 \)?
3. How does the sample size influence the test statistic and margin of error?
4. What are the assumptions for performing a hypothesis test on the population mean?
5. How would the result change if the sample mean were higher than 8.17 ounces?

**Tip**: In hypothesis testing, always define whether you're doing a one-tailed or two-tailed test before calculating any values.

A manufacturer of chocolate candies uses machines to package candies as they move along a filling line. Although the packages are labeled as 8 ounces, the company wants the packages to contain a mean of 8.17 ounces so that virtually none of the packages contain less than 8 ounces. A sample of 50 packages is selected periodically, and the packaging process is stopped if there is evidence that the mean amount packaged is different from 8.17 ounces. Suppose that in a particular sample of 50 packages, the mean amount dispensed is 8.165 ounces, with a sample standard deviation of 0.046 ounce. Is there evidence that the population mean amount is different from 8.17 ounces? (Use a 0.05 level of significance.) State the null and alternative hypotheses.

Solve a hypothesis test problem on whether the mean weight of chocolate candy packages is different from the target mean of 8.17 ounces. This problem involves key statistical concepts like hypothesis testing, two-tailed tests, and standard error calculations. The solution uses a sample size of 50 and a standard deviation of 0.046 ounces, with a significance level of 0.05.

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