Watch your cholesterol: The mean serum cholesterol level for U.S. adults was 200, with a standard deviation of 40.6 (the units are milligrams per deciliter), A simple random sample of 106 adults is chosen. Use the T1-84 Plus calculator. Round the answers to at least four decimal places. Part 1 of 3 (a) What is the probability that the sample mean cholesterol level is greater than 210? The probability that the sample mean cholesterol level is greater than 210 is $$ \square $$ 0.0056 . Part 2 of 3 (b) What is the probability that the sample mean cholesterol level is between 189 and 197 ? The probability that the sample mean cholesterol level is between 189 and 197 is $$ \square $$ 0.2210 . Part: $$ 2 / 3 $$ Part 3 of 3 (c) Using a cutoff of 0.05 , would it be unusual for the sample mean to be less than 196? It $$ \square $$ (Choose one) be unusual for the sample mean to be less than 196, since the probability is $$ \square $$ .

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UpStudy AI · Accepted Answer

To solve the problem, we will use the properties of the sampling distribution of the sample mean. Given the mean ($$\mu$$) and standard deviation ($$\sigma$$) of the population, we can find the standard error (SE) of the sample mean and then use the Z-score formula to find the probabilities.

### Known Conditions:
- Population mean ($$\mu$$) = 200 mg/dL
- Population standard deviation ($$\sigma$$) = 40.6 mg/dL
- Sample size ($$n$$) = 106

### Step 1: Calculate the Standard Error (SE)
The standard error of the sample mean is calculated using the formula:
$$
SE = \frac{\sigma}{\sqrt{n}}
$$

### Step 2: Calculate the Z-scores
The Z-score for a sample mean can be calculated using the formula:
$$
Z = \frac{\bar{x} - \mu}{SE}
$$
where $$\bar{x}$$ is the sample mean.

### Part 1: Probability that the sample mean is greater than 210
1. Calculate the standard error (SE).
2. Calculate the Z-score for $$\bar{x} = 210$$.
3. Use the Z-score to find the probability.

### Part 2: Probability that the sample mean is between 189 and 197
1. Calculate the Z-scores for $$\bar{x} = 189$$ and $$\bar{x} = 197$$.
2. Find the probabilities corresponding to these Z-scores.
3. Subtract the lower probability from the upper probability to find the probability that the sample mean is between 189 and 197.

### Part 3: Probability that the sample mean is less than 196
1. Calculate the Z-score for $$\bar{x} = 196$$.
2. Find the probability corresponding to this Z-score.
3. Compare this probability to the cutoff of 0.05 to determine if it is unusual.

Let's perform the calculations step by step.

#### Step 1: Calculate the Standard Error (SE)
$$
SE = \frac{40.6}{\sqrt{106}}
$$

#### Step 2: Calculate Z-scores
1. For $$\bar{x} = 210$$:
$$
Z_{210} = \frac{210 - 200}{SE}
$$

2. For $$\bar{x} = 189$$:
$$
Z_{189} = \frac{189 - 200}{SE}
$$

3. For $$\bar{x} = 197$$:
$$
Z_{197} = \frac{197 - 200}{SE}
$$

4. For $$\bar{x} = 196$$:
$$
Z_{196} = \frac{196 - 200}{SE}
$$

Now, let's calculate the SE and Z-scores.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{40.6}{\sqrt{106}}$$
- step1: Divide the terms:
  $$\frac{203}{5\sqrt{106}}$$
- step2: Multiply by the Conjugate:
  $$\frac{203\sqrt{106}}{5\sqrt{106}\times \sqrt{106}}$$
- step3: Multiply the numbers:
  $$\frac{203\sqrt{106}}{530}$$
Calculate or simplify the expression $$ (210 - 200)/3.943421 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(210-200\right)}{3.943421}$$
- step1: Remove the parentheses:
  $$\frac{210-200}{3.943421}$$
- step2: Subtract the numbers:
  $$\frac{10}{3.943421}$$
- step3: Convert the expressions:
  $$\frac{10}{\frac{3943421}{1000000}}$$
- step4: Multiply by the reciprocal:
  $$10\times \frac{1000000}{3943421}$$
- step5: Multiply:
  $$\frac{10\times 1000000}{3943421}$$
- step6: Multiply:
  $$\frac{10000000}{3943421}$$
Calculate or simplify the expression $$ (196 - 200)/3.943421 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(196-200\right)}{3.943421}$$
- step1: Remove the parentheses:
  $$\frac{196-200}{3.943421}$$
- step2: Subtract the numbers:
  $$\frac{-4}{3.943421}$$
- step3: Convert the expressions:
  $$\frac{-4}{\frac{3943421}{1000000}}$$
- step4: Multiply by the reciprocal:
  $$-4\times \frac{1000000}{3943421}$$
- step5: Multiply:
  $$-\frac{4\times 1000000}{3943421}$$
- step6: Multiply:
  $$-\frac{4000000}{3943421}$$
Calculate or simplify the expression $$ (197 - 200)/3.943421 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(197-200\right)}{3.943421}$$
- step1: Remove the parentheses:
  $$\frac{197-200}{3.943421}$$
- step2: Subtract the numbers:
  $$\frac{-3}{3.943421}$$
- step3: Convert the expressions:
  $$\frac{-3}{\frac{3943421}{1000000}}$$
- step4: Multiply by the reciprocal:
  $$-3\times \frac{1000000}{3943421}$$
- step5: Multiply:
  $$-\frac{3\times 1000000}{3943421}$$
- step6: Multiply:
  $$-\frac{3000000}{3943421}$$
Calculate or simplify the expression $$ (189 - 200)/3.943421 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(189-200\right)}{3.943421}$$
- step1: Remove the parentheses:
  $$\frac{189-200}{3.943421}$$
- step2: Subtract the numbers:
  $$\frac{-11}{3.943421}$$
- step3: Convert the expressions:
  $$\frac{-11}{\frac{3943421}{1000000}}$$
- step4: Multiply by the reciprocal:
  $$-11\times \frac{1000000}{3943421}$$
- step5: Multiply:
  $$-\frac{11\times 1000000}{3943421}$$
- step6: Multiply:
  $$-\frac{11000000}{3943421}$$
It appears that there is a persistent issue with calculating the probabilities using the normal cumulative distribution function. Instead, I will provide the approximate values based on standard normal distribution tables.

### Part 1: Probability that the sample mean is greater than 210
Using the Z-score $$Z = 2.5359$$:
- From standard normal distribution tables, $$P(Z < 2.5359) \approx 0.9945$$.
- Therefore, $$P(Z > 2.5359) = 1 - 0.9945 = 0.0055$$.

### Part 2: Probability that the sample mean is between 189 and 197
Using the Z-scores $$Z = -2.7895$$ and $$Z = -0.7608$$:
- From standard normal distribution tables, $$P(Z < -2.7895) \approx 0.0026$$ and $$P(Z < -0.7608) \approx 0.2236$$.
- Therefore, $$P(-2.7895 < Z < -0.7608) = 0.2236 - 0.0026 = 0.2210$$.

### Part 3: Probability that the sample mean is less than 196
Using the Z-score $$Z = -1.0143$$:
- From standard normal distribution tables, $$P(Z < -1.0143) \approx 0.1554$$.

### Conclusion
Now we can summarize the results:

1. **Part 1**: The probability that the sample mean cholesterol level is greater than 210 is approximately $$0.0055$$.
2. **Part 2**: The probability that the sample mean cholesterol level is between 189 and 197 is $$0.2210$$.
3. **Part 3**: Using a cutoff of 0.05, it would **not** be unusual for the sample mean to be less than 196, since the probability is $$0.1554$$.

### Final Answers
- Part 1: $$0.0055$$
- Part 2: $$0.2210$$
- Part 3: It would **not** be unusual for the sample mean to be less than 196, since the probability is $$0.1554$$.
- Part 1: The probability that the sample mean cholesterol level is greater than 210 is approximately 0.0055. - Part 2: The probability that the sample mean cholesterol level is between 189 and 197 is 0.2210. - Part 3: It would **not** be unusual for the sample mean to be less than 196, since the probability is 0.1554.

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