According to “Dietary Goals for the United States (1997)”, high sodium intake may be related to ulcers, stomach cancer, and migraine headaches. The human requirement for salt is only 220 milligram per day, which is surpassed in most single servings of ready-to-eat cereals. If a random sample of 20 similar servings of Special K has a mean sodium content of 244 milligrams and a standard deviation 24,5 milligrams does this suggest at 0,05 level of significance that the average sodium content for single servings of Special K is greater than 220 milligrams? Assume the distribution of sodium content to be normal. State the null and alternative hypothesis.
Null hypothesis (H0): The average sodium content for single servings of Special K is not greater than 220 milligrams.
Alternative hypothesis (H1): The average sodium content for single servings of Special K is greater than 220 milligrams.
Null hypothesis (H0): The average sodium content for single servings of Special K is equal to or less than 220 milligrams.
Alternative hypothesis (H1): The average sodium content for single servings of Special K is greater than 220 milligrams.
To determine whether the average sodium content for single servings of Special K is greater than 220 milligrams, we can use a hypothesis test. The first step is to state the null and alternative hypotheses:
Null hypothesis (H0): The average sodium content for single servings of Special K is equal to or less than 220 milligrams.
Alternative hypothesis (HA): The average sodium content for single servings of Special K is greater than 220 milligrams.
In mathematical notation:
H0: μ ≤ 220
HA: μ > 220
Where:
H0 represents the null hypothesis
HA represents the alternative hypothesis
μ represents the population mean sodium content for single servings of Special K
Next, we'll perform the hypothesis test using the sample data provided.
The significance level given is 0.05, which means we are willing to accept a 5% chance of rejecting the null hypothesis when it is true (Type I error).
To perform the hypothesis test, we'll calculate the test statistic and compare it to the critical value of the t-distribution corresponding to the significance level and degrees of freedom (n - 1).
The test statistic is given by:
t = (x̄ - μ) / (s / √n)
Where:
x̄ represents the sample mean sodium content (244 milligrams)
μ represents the population mean sodium content (220 milligrams)
s represents the sample standard deviation (24.5 milligrams)
n represents the sample size (20)
Calculating the test statistic:
t = (244 - 220) / (24.5 / √20)
We'll also calculate the degrees of freedom for our t-distribution, which is (n - 1):
df = 20 - 1 = 19
The critical value for a one-tailed test with a significance level of 0.05 and 19 degrees of freedom can be found using a t-distribution table or a statistical software.
Finally, we'll compare our test statistic to the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the average sodium content for single servings of Special K is greater than 220 milligrams.
Note: The calculations to find the critical value and compare it to the test statistic are not provided here, but they can be easily done using statistical software or a t-distribution table.