Question 1183827: Q1. A researcher claims that the yearly consumption of soft drinks per person is 52 gallons. In a sample of 50 randomly selected people, the mean of the yearly consumption was 56.3 gallons. The standard deviation of the population is 3.5 gallons. Use this information to answer the following questions
a. State the null and the alternative hypothesis if a statics test is to be conducted
b. Find the absolute difference between the degrees of freedom and the sample size.
c. If in the sample of 50, only 8 people do not consume soft drink, compute the product of the proportion of those who consumed the soft drink and those who do not consume the soft drink.
d. What is the total yearly consumption of the 50 randomly selected people?
e. Compute the test statistics form the information given.
f. What is p - value for the test?
g. Based on the P-value, is the researcher’s claim valid at 0.05 significant level?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the solution:
**a. Null and Alternative Hypotheses:**
* **Null Hypothesis (H0):** The yearly consumption of soft drinks per person is 52 gallons. (μ = 52)
* **Alternative Hypothesis (H1):** The yearly consumption of soft drinks per person is *not* 52 gallons. (μ ≠ 52) This is a two-tailed test.
**b. Difference between Degrees of Freedom and Sample Size:**
* Degrees of freedom (df) = Sample size (n) - 1 = 50 - 1 = 49
* Absolute difference = |n - df| = |50 - 49| = 1
**c. Product of Proportions:**
* People who consumed soft drinks = 50 - 8 = 42
* Proportion who consumed soft drinks = 42/50 = 0.84
* Proportion who did not consume soft drinks = 8/50 = 0.16
* Product of proportions = 0.84 * 0.16 = 0.1344
**d. Total Yearly Consumption:**
* Total consumption = Sample mean * Sample size
* Total consumption = 56.3 gallons/person * 50 people = 2815 gallons
**e. Test Statistic:**
Since the population standard deviation is known, we use a z-test.
z = (sample mean - population mean) / (population standard deviation / √sample size)
z = (56.3 - 52) / (3.5 / √50)
z = 4.3 / (3.5 / 7.071)
z ≈ 4.3 / 0.495
z ≈ 8.69
**f. P-value:**
Because this is a two-tailed test, we need to find the area in *both* tails of the standard normal distribution that is beyond our calculated z-score.
* Since the z-score is very large (8.69), the p-value will be extremely small, essentially close to zero. Most z-tables won't go out this far; a calculator or statistical software is best.
**g. Validity of the Researcher's Claim:**
* Significance level (α) = 0.05
* Our p-value is essentially 0, which is *much* less than 0.05.
Since the p-value is less than the significance level, we reject the null hypothesis.
**Conclusion:** There is extremely strong evidence to suggest that the researcher's claim is *not* valid. The yearly consumption of soft drinks per person is significantly different from 52 gallons. The sample data suggests it is higher.
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