A coffee company wants to make sure that there coffee is being served at the right temperature. If it is too hot, the customers could burn themselves. If it is too cold, the customers will be unsatisfied. The company has determined that they want the average coffee temperature to be 65 degrees C. They take a sample of 20 orders of coffee and find the sample mean to be equal to 70.2 C.
What does mu represent for this problem?
mu = mean (typically of the population)
To determine whether the coffee is being served at the right temperature, we can perform a hypothesis test using the sample data. Let's follow these steps:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1):
- The null hypothesis (H0) states that the average coffee temperature is equal to 65 degrees C.
- The alternative hypothesis (H1) states that the average coffee temperature is not equal to 65 degrees C.
Step 2: Set the significance level (α):
- The significance level is the threshold beyond which we will reject the null hypothesis if the evidence is strong enough. Let's assume a significance level of 0.05, which is commonly used.
Step 3: Conduct the test and calculate the test statistic:
- We can use a t-test since the population standard deviation is not known.
- The test statistic (t-score) can be calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / √sample size)
Given:
- Sample mean (x̄) = 70.2 degrees C
- Hypothesized mean (μ) = 65 degrees C
- Sample size (n) = 20
To find the sample standard deviation (s), we need additional information. Please provide the sample standard deviation or the raw data values if available.
Once we have the sample standard deviation (s), we can calculate the test statistic (t-score) using the given formula.
To determine whether the coffee is being served at the right temperature, we can use hypothesis testing. Specifically, we can perform a one-sample t-test to compare the sample mean to the desired temperature.
Here's how you can perform the t-test to determine the significance of the temperature difference:
Step 1: Set up the hypotheses
The null hypothesis (H0): The average coffee temperature is equal to 65 degrees C.
The alternative hypothesis (Ha): The average coffee temperature is not equal to 65 degrees C.
Step 2: Choose the significance level
The significance level (alpha) is the probability of rejecting the null hypothesis when it is true. Let's say the significance level is 0.05, which is a common choice.
Step 3: Compute the test statistic
To calculate the test statistic, we need the sample mean, sample standard deviation, sample size, and the hypothesized population mean.
In this case:
Sample mean (x̄) = 70.2 C
Hypothesized population mean (μ0) = 65 C
Sample size (n) = 20
Step 4: Calculate the standard error
The standard error (SE) represents the average amount of variability in the sample mean from sample to sample. In this case, you can calculate the standard error using the formula:
SE = sample standard deviation / √(sample size)
Since we don't have the sample standard deviation, we'll use the sample standard deviation formula:
s = √(Σ(x - x̄)² / (n - 1))
You'll need to calculate the sample standard deviation 's' using the given data and then use it to calculate the standard error 'SE.'
Step 5: Calculate the t-statistic
Using the formula:
t = (x̄ - μ0) / SE
Replace 'x̄' with the sample mean, 'μ0' with the desired mean, and 'SE' with the standard error calculated in step 4.
Step 6: Determine the p-value
The p-value is the probability of obtaining a sample mean as extreme as the one observed in the data, assuming the null hypothesis is true. We can determine the p-value by finding the area under the t-distribution curve corresponding to the test statistic 't.' This can be done using statistical software or looking up critical values in a t-distribution table.
Step 7: Make a decision
Compare the obtained p-value to the significance level. If the p-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
By following these steps, you can determine whether the coffee is being served at the right temperature based on the sample mean of 70.2 C and the desired temperature of 65 C.