Part 1 Of 3: Question 1 Of 2010 Points In An Article

Part 1 Of 3 Question 1 Of 2010 Pointsin An Article Appearing Intoday

In an article appearing in Today’s Health, a writer states that the average number of calories in a serving of popcorn is 75. To determine if the average number of calories in a serving of popcorn is different from 75, a nutritionist selected a random sample of 20 servings of popcorn, which had a sample mean of 78 calories and a standard deviation of 7. The significance level for the test is α = 0.05. The goal is to test whether there is enough evidence to reject the writer's claim about the average calories in a serving of popcorn.

Paper For Above instruction

The claim about the average number of calories in a serving of popcorn can be examined through hypotheses testing, specifically a two-tailed t-test since the population standard deviation is unknown, and the sample size is relatively small (n=20). The null hypothesis (H₀) posits that the true population mean is equal to 75 calories, the value claimed by the article. In contrast, the alternative hypothesis (H₁) asserts that the mean is different from 75, acknowledging the possibility of either an increase or decrease in calories per serving.

Formally, these hypotheses are expressed as:

  • H₀: μ = 75
  • H₁: μ ≠ 75

Given the sample mean (x̄) of 78, sample standard deviation (s) of 7, and sample size (n) of 20, the test statistic is calculated using the t-distribution:

t = (x̄ - μ₀) / (s/√n)

Substituting the values:

t = (78 - 75) / (7 / √20) ≈ 3 / (7 / 4.4721) ≈ 3 / 1.565 ≈ 1.917

The degrees of freedom for this test are n - 1 = 19. Using a t-distribution table or software, the critical t-value for a two-tailed test at α = 0.05 and df = 19 is approximately ±2.093. Since the calculated t-value (≈1.917) is within the range (-2.093, 2.093), we do not reject the null hypothesis at the 5% significance level.

The p-value corresponding to the test statistic can be obtained from statistical software or t-tables. It is approximately 0.07, which exceeds the significance level of 0.05. Thus, there is insufficient evidence to reject the claim that the average number of calories per serving of popcorn is 75. In other words, the data do not provide enough evidence to conclude a significant difference from 75 calories.

In conclusion, based on this sample, we fail to reject the null hypothesis. The sample data are consistent with the claim in the article that a typical serving contains about 75 calories, at the 5% significance level. This implies that any observed difference might be due to random variation, and there is no strong evidence to assert a change in the average caloric content per serving.

References

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