(b) the figure below is a histogram of 200 diameters obtained by using one of the two sampling methods described. considering the shape of the histogram, explain which method, method 1 or method 2, was most likely used to obtain a such a sample.

(c) which of the two sampling methods, method 1 or method 2, will result in less variability in the diameters of the 200 tortillas in the sample on a given day? explain.

each day, the distribution of the 200,000 tortillas made that day has mean diameter 6 inches with standard deviation 0.11 inch.

(d) for samples of size 200 taken from one day’s production, describe the sampling distribution of the sample mean diameter for samples that are obtained using method 1.

(e) suppose that one of the two sampling methods will be selected and used every day for one year (365 days). the sample mean of the 200 diameters will be recorded each day. which of the two methods will result in less variability in the distribution of the 365 sample means? explain.

(f) a government inspector will visit the facility on june 22 to observe the sampling and to determine if the factory is in compliance with the advertised mean diameter of 6 inches. the manager knows that, with both sampling methods, the sample mean is an unbiased estimator of the population mean. however, the manager is unsure which method is more likely to produce a sample mean that is close to 6 inches on the day of sampling. based on your previous answers, which of the two sampling methods, method 1 or method 2, is more likely to produce a sample mean close to 6 inches? explain.

Question

(b) the figure below is a histogram of 200 diameters obtained by using one of the two sampling methods described. considering the shape of the histogram, explain which method, method 1 or method 2, was most likely used to obtain a such a sample.

(c) which of the two sampling methods, method 1 or method 2, will result in less variability in the diameters of the 200 tortillas in the sample on a given day? explain.

each day, the distribution of the 200,000 tortillas made that day has mean diameter 6 inches with standard deviation 0.11 inch.

(d) for samples of size 200 taken from one day’s production, describe the sampling distribution of the sample mean diameter for samples that are obtained using method 1.

(e) suppose that one of the two sampling methods will be selected and used every day for one year (365 days). the sample mean of the 200 diameters will be recorded each day. which of the two methods will result in less variability in the distribution of the 365 sample means? explain.

(f) a government inspector will visit the facility on june 22 to observe the sampling and to determine if the factory is in compliance with the advertised mean diameter of 6 inches. the manager knows that, with both sampling methods, the sample mean is an unbiased estimator of the population mean. however, the manager is unsure which method is more likely to produce a sample mean that is close to 6 inches on the day of sampling. based on your previous answers, which of the two sampling methods, method 1 or method 2, is more likely to produce a sample mean close to 6 inches? explain.

Accepted Answer

### Part (b)
# Brief Explanations:
To determine the sampling method, we analyze the histogram's shape. If Method 1 samples randomly (e.g., across production), it might have more variability, while Method 2 (e.g., from one machine) could have a more clustered or bimodal? Wait, the histogram has two main clusters? Wait, the population mean is 6. The histogram has two peaks? Wait, maybe Method 2 samples from a single machine (consistent process), but the histogram here has two main groups? Wait, no—maybe Method 1 is random sampling (so captures the overall distribution with two modes if production has two machines), and Method 2 is from one machine (single mode). Wait, the histogram has two peaks: around 5.85 - 5.925 and 6.075 - 6.15. So if Method 1 samples from all machines (multiple processes), it would show multiple modes, while Method 2 (one machine) would have one mode. So the histogram with two modes suggests Method 1 (sampling across machines) was used, as it captures different machine outputs.
# Answer:
Method 1 was most likely used. The histogram has two distinct clusters (modes), suggesting sampling from multiple sources (e.g., different machines) with varying diameters, consistent with a random sampling method (Method 1) that captures the overall population’s variability, unlike Method 2 (e.g., sampling from one machine, which would likely have a single - mode distribution).

### Part (c)
# Brief Explanations:
Variability in samples: Method 2 (e.g., sampling from one machine) has less variability because the tortillas from a single machine are more consistent (less variation in production process for one machine) compared to Method 1 (sampling from multiple machines, which may have different diameter distributions, increasing overall sample variability).
# Answer:
Method 2 will result in less variability. If Method 2 samples from a single production source (e.g., one machine), the tortillas’ diameters are more consistent (less variation in the production process of one machine). Method 1 (sampling from multiple sources) includes diameters from different, potentially variable, production processes, increasing the sample’s variability.

### Part (d)
# Brief Explanations:
For sampling distribution of the sample mean (size $ n = 200 $): By the Central Limit Theorem (CLT), if the population is any distribution (or normal), the sampling distribution of the sample mean $\bar{x}$ is approximately normal. The population mean $\mu=6$ inches, standard deviation $\sigma = 0.11$ inches. The mean of the sampling distribution is $\mu_{\bar{x}}=\mu = 6$ inches. The standard deviation (standard error) is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{0.11}{\sqrt{200}}\approx\frac{0.11}{14.142}\approx0.0078$ inches. So the sampling distribution is approximately normal with mean 6 inches and standard deviation $\approx0.0078$ inches.
# Answer:
The sampling distribution of the sample mean diameter (for Method 1) is approximately normal (by the Central Limit Theorem, since $ n = 200 $ is large) with mean $\boldsymbol{\mu_{\bar{x}} = 6}$ inches (equal to the population mean) and standard deviation (standard error) $\boldsymbol{\sigma_{\bar{x}}=\frac{0.11}{\sqrt{200}}\approx0.0078}$ inches.

### Part (e)
# Brief Explanations:
Variability of sample means (sampling distribution standard error): The standard error is $\frac{\sigma}{\sqrt{n}}$, but also, if Method 2 has less variability in the sample (from part c), but for the sampling distribution of the sample mean, the standard error depends on the population standard deviation and sample size. Wait, no—both methods sample from the same population (daily production, $\sigma = 0.11$). But if Method 2’s samples have less variability (smaller sample standard deviation $ s $), but the sampling distribution of the sample mean’s standard error is $\frac{\sigma}{\sqrt{n}}$ (for population $\sigma$) or $\frac{s}{\sqrt{n}}$ (for sample $ s $). However, since both methods use $ n = 200 $, and the population $\sigma$ is fixed, but if Method 2’s samples have less variability (smaller $ s $), its sampling distribution of the sample mean would have less variability? Wait, no—actually, the sampling distribution’s standard error for the sample mean is determined by the population standard deviation (if known) or sample standard deviation. But since both methods sample from the same daily population (mean 6, $\sigma = 0.11$), the standard error for the sample mean is $\frac{\sigma}{\sqrt{n}}$ for both. But wait, no—if Method 2’s samples have less variability (smaller $ s $), then the standard error of its sample mean would be smaller. Wait, the key is: The variability of the sample means (sampling distribution) is measured by the standard error. For Method 2, the samples have less variability (smaller $ s $), so when we take the sample mean, the standard error $\frac{s}{\sqrt{n}}$ would be smaller. Alternatively, if we consider the population standard deviation, but the daily population is fixed. Wait, the question is about the distribution of the 365 sample means. The method with less variability in the sample (Method 2, from part c) will have sample means with less variability, because the sample mean’s variability is related to the sample’s variability. So Method 2, with less variable samples, will have sample means with less variability (smaller standard error in their sampling distribution of sample means).
# Answer:
Method 2 will result in less variability in the distribution of the 365 sample means. Method 2 produces samples with less variability in tortilla diameters (part c). The variability of the sample mean is related to the sample’s variability: a less variable sample (smaller sample standard deviation $ s $) leads to a less variable sampling distribution of the sample mean (smaller standard error $\frac{s}{\sqrt{n}}$), so over 365 days, Method 2’s sample means will be less variable.

### Part (f)
# Brief Explanations:
To be close to 6 inches, we want a method with less variability in the sample mean (so the sample mean is more likely to be near the population mean). From part (e), Method 2 has less variability in the sample means’ distribution (smaller standard error), meaning its sample means are more concentrated around the population mean (6 inches). Also, since both methods are unbiased, the method with less variability in the sample mean’s distribution is more likely to produce a sample mean close to 6 inches.
# Answer:
Method 2 is more likely to produce a sample mean close to 6 inches. Method 2 has less variability in the sample (part c) and thus in the sampling distribution of the sample mean (part e). A less variable sampling distribution (smaller standard error) means the sample mean is more concentrated around the population mean (6 inches), making it more likely to be close to 6 inches.

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Question

Explanation:

Part (b)

Answer:

Part (c)