a manufacturer of detergent claims that the contents of the boxes in the market weigh on average at least 16 oz.. the distribution of weights is known to be normal with a standard deviation of 0.4 oz. to test the claim of the manufacturer a random sample of 16 boxes were weighed and it was found that the sample mean weight is 15.84 oz. test at levels of significance of 5%, 10% and the null hypothesis that the population mean weight is at least 16 oz.

To test the null hypothesis that the population mean weight is at least 16 oz, we can use a one-sample t-test. Here's how you can perform the test at the given significance levels:

Step 1: State the null and alternative hypothesis.
- Null hypothesis (H0): The population mean weight is at least 16 oz.
- Alternative hypothesis (Ha): The population mean weight is less than 16 oz.

Step 2: Determine the level of significance (α).
- For the 5% significance level, α = 0.05.
- For the 10% significance level, α = 0.10.

Step 3: Calculate the test statistic.
- The test statistic for a one-sample t-test is given by: t = (x̄ - μ) / (s / √n)
- x̄ represents the sample mean weight (in this case, 15.84 oz).
- μ represents the population mean weight (16 oz).
- s represents the standard deviation of the population (0.4 oz).
- n represents the sample size (16 boxes).

Step 4: Determine the critical value.
- The critical value is the value at which we reject the null hypothesis.
- For a one-tailed test at a given significance level, we determine the critical value using a t-distribution table or a statistical software.
- For the 5% level of significance, t_critical = -1.753.
- For the 10% level of significance, t_critical = -1.340.

Step 5: Compare the test statistic with the critical value and make a decision.
- If the test statistic is less than the critical value, we reject the null hypothesis; otherwise, we fail to reject it.

Let's calculate the test statistic:

t = (15.84 - 16) / (0.4 / √16)
t = -0.16 / (0.4 / 4)
t = -0.16 / 0.1
t = -1.6

Now, let's compare the test statistic with the critical values:

At the 5% significance level:
t_critical = -1.753
Since t = -1.6 > -1.753, we fail to reject the null hypothesis.

At the 10% significance level:
t_critical = -1.340
Since t = -1.6 > -1.340, we also fail to reject the null hypothesis.

Therefore, based on our calculations, we do not have enough evidence to reject the manufacturer's claim that the average weight of the detergent boxes is at least 16 oz at both the 5% and 10% levels of significance.