We Want To Test A Manufacturer's Claim About The Average Lif

We Want To Test A Manufacturers Claim That The Average Life Of Its Pr

We want to test a manufacturer's claim that the average life of its product is 100 hours (H0: µ = 100). There is evidence that the true average life of the product is not the advertised claim. A hypothesis test was carried out on data collected on the life of the product with the details given below. Which of the following significance levels would lead us to reject the manufacturer's claim? Test of mu = 100 vs not = 100, with p-value = 0.%, 10%, 1%, or 2% ?

Seafood restaurants along the coast of New England offer a variety of entrees, but lobster is the most popular meal. At Newick’s Lobster Home in Dover, New Hampshire, 37% of all diners order lobster. Suppose 120 customers are selected at random, and the meal ordered by each is recorded. Fill in the blank. (Give your answer to four decimal places.) The probability that the sample proportion of diners who order lobster is less than 0.30 is . The manager of Newick’s is concerned that more customers might be ordering lobster. This would require a change in restaurant ordering and a shift in kitchen staff. Suppose the actual proportion of diners who order lobster is 0.43. Is there any evidence to suggest that the proportion of diners who order lobster has increased? yes or no?

Paper For Above instruction

The purpose of this paper is to analyze two statistical scenarios involving hypothesis testing related to the claims about a product’s average lifespan and customers’ ordering habits in a restaurant. The first scenario tests a manufacturer’s claim regarding the average lifespan of its product, while the second involves evaluating whether the proportion of diners ordering lobsters has increased over time. Both cases utilize hypothesis testing principles to interpret data and make informed decisions based on significance levels and p-values.

Analysis of the Manufacturer’s Claim about Product Life Expectancy

The manufacturer asserts that the mean life of its product is 100 hours. To test this claim, a hypothesis testing approach is implemented. The null hypothesis (H0: μ = 100) assumes the manufacturer’s claim is correct, while the alternative hypothesis (H1: μ ≠ 100) suggests the true mean differs from 100 hours. The data collected from testing provides a p-value, which quantifies the probability of observing the data assuming H0 is true.

Given a p-value (for instance, 0.05 or 5%), decision-making depends on the chosen significance level (α). Common thresholds include 10%, 2%, 1%, and 0.5%. If the p-value is less than or equal to α, the evidence is sufficient to reject H0 at that level.

In this scenario, the p-value is indicated as 0.%, which appears to be incomplete. Assuming typical p-value ranges, if the p-value were less than 1% (0.01), then at significance levels of 1% or higher (like 2%, 10%, 0.05), we would reject the manufacturer’s claim. Conversely, if the p-value exceeds the significance level, we fail to reject H0.

Therefore, the significance levels at which the test would lead us to reject the claim are those exceeding the p-value threshold, with the lowest being 1%—a standard criterion for strict decision-making in hypothesis testing.

Survey of Lobster Ordering Behavior

The second scenario involves a survey of diners at Newick’s Lobster Home in Dover, New Hampshire, where 37% of all diners typically order lobster. In a sample of 120 customers, the manager is interested in understanding whether the proportion of diners who order lobster has decreased below 30% or increased from the prior 37%.

The probability that the sample proportion of diners who order lobster is less than 0.30 is to be calculated. This involves understanding the sampling distribution of the sample proportion (p̂). Under the null hypothesis, assuming the true proportion p = 0.37, the sampling distribution of p̂ is approximately normal for sufficiently large samples, with mean p and standard deviation sqrt[p(1-p)/n].

Calculating this probability entails evaluating P(p̂

Z = (0.30 - 0.37) / sqrt[0.37 × 0.63 / 120]

Applying this formula yields the probability to four decimal places, indicating the likelihood that the sample proportion falls below 0.30 purely by chance if p = 0.37.

Subsequently, the question of whether the proportion of diners ordering lobster has increased from 0.37 to 0.43 is addressed. Under the null hypothesis H0: p = 0.37, testing against H1: p > 0.43, involves calculating a Z-test statistic. If the data shows that this statistic exceeds the critical value at a chosen significance level, or equivalently, if the p-value corresponding to the observed statistic is below the significance threshold, then there is statistically significant evidence suggesting an increase in the proportion of lobster orders.

Given the increase from 0.37 to 0.43, if the p-value is small enough (less than the significance level, such as 0.05), this indicates evidence supporting the claim that more customers are ordering lobster.

In conclusion, hypothesis testing provides a structured methodology to evaluate claims based on sample data. For the manufacturer, the significance level determines when the claim about product life is rejected, while for the restaurant, calculating probabilities and p-values informs decisions about changing operations based on diner preferences.

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