Question 1170482
Let's perform the hypothesis test and calculate the confidence interval.

**1. Define Hypotheses**

* **Null Hypothesis (H0):** The mean weight of sugar packages is 500 grams (μ = 500).
* **Alternative Hypothesis (H1):** The mean weight of sugar packages is less than 500 grams (μ < 500).

**2. Set Significance Level (α)**

* α = 0.10

**3. Determine Degrees of Freedom (df)**

* Sample size (n) = 20
* df = n - 1 = 20 - 1 = 19

**4. Find the Critical t-value**

* This is a left-tailed test since H1: μ < 500.
* Using a t-distribution table or calculator with α = 0.10 and df = 19, the critical t-value is approximately -1.729.

**5. Compute the t-statistic**

* Sample mean (x̄) = 496 grams
* Sample standard deviation (s) = 8 grams
* Population mean (μ) = 500 grams
* t = (x̄ - μ) / (s / √n)
* t = (496 - 500) / (8 / √20)
* t = -4 / (8 / 4.472)
* t = -4 / 1.789
* t ≈ -2.236

**6. Decision Rule**

* Reject H0 if the calculated t-statistic is less than the critical t-value (-1.729).

**7. Conclusion**

* Since the calculated t-statistic (-2.236) is less than the critical t-value (-1.729), we reject the null hypothesis.
* Therefore, we conclude that there is sufficient evidence at the 10% significance level to suggest that the mean weight of the sugar packages is less than 500 grams.

**8. Compute the 90% Confidence Interval**

* Confidence level = 90%
* α = 1 - 0.90 = 0.10
* α / 2 = 0.05
* Using a t-distribution table or calculator with α/2 = 0.05 and df = 19, the t-value is approximately 1.729.
* Confidence interval = x̄ ± t * (s / √n)
* Confidence interval = 496 ± 1.729 * (8 / √20)
* Confidence interval = 496 ± 1.729 * 1.789
* Confidence interval = 496 ± 3.093
* Confidence interval = (492.907, 499.093)

**Summary**

* **Hypothesis:**
    * H0: μ = 500
    * H1: μ < 500
* **Level of significance:** α = 0.10
* **Degrees of freedom:** df = 19
* **Critical region:** t < -1.729
* **t-statistic:** t ≈ -2.236
* **Decision rule:** Reject H0 if t < -1.729.
* **Conclusion:** Reject H0. The mean weight is less than 500 grams.
* **90% Confidence Interval:** (492.907, 499.093)