You would like to construct a 99% confidence interval to estimate the population mean time it takes drivers to react following the application of brakes by the driver in front of them. You take a random sample of reaction time measurements and compute their mean to be 1.8 seconds and their standard deviation to be 0.4 seconds.A. What is the best point estimate, based on the sample, to use for the population mean? (Answer is in seconds)A=B. For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 99% confidence interval for the population mean. (See picture) ### Sampling Scenarios: Choosing the Appropriate Statistical TestBelow is a table summarizing various sampling scenarios and the appropriate tests to use (Z or t). The scenarios take into consideration the sample size, distribution, and whether the standard deviation is known.| **Sampling scenario** | **Z** | **t** | **Could use either $ Z $ or $ t $** | **Unclear** ||:----------------------|:-----:|:-----:|:---------------------------------------:|:-----------:|| The sample has size 10, and it is from a normally distributed population with an unknown standard deviation. | ⃝ | ⃝ | ⃝ | ⃝ || The sample has size 90, and it is from a non-normally distributed population with a known standard deviation of 0.45. | ⃝ | ⃝ | ⃝ | ⃝ || The sample has size 85, and it is from a non-normally distributed population. | ⃝ | ⃝ | ⃝ | ⃝ |### Explanation- **First Scenario**: - **Sampling context**: Small sample size (n=10) from a normally distributed population with an unknown standard deviation. - **Test choice**: Typically, for small sample sizes and unknown standard deviations from normally distributed populations, the t-test is more appropriate.- **Second Scenario**: - **Sampling context**: Large sample size (n=90) from a non-normally distributed population with a known standard deviation (0.45). - **Test choice**: For large sample sizes, the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal. Thus, either Z or t tests could potentially be used, but the Z-test might be preferred due to the known standard deviation.- **Third Scenario**: - **Sampling context**: Large sample size (n=85) from a non-normally distributed population without specification of the standard deviation. - **Test choice**: Similar to the second scenario, a large sample size means the Central Limit Theorem can be applied, and thus either test can often be used.The circles (⃝) are placeholders indicating the options evaluated in each scenario. For educational purposes, each scenario should be further analyzed to determine the most precise statistical test based on given conditions

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You would like to construct a 99% confidence interval to estimate the population mean time it takes drivers to react following the application of brakes by the driver in front of them. You take a random sample of reaction time measurements and compute their mean to

You would like to construct a 99% confidence interval to estimate the population mean time it takes drivers to react following the application of brakes by the driver in front of them. You take a random sample of reaction time measurements and compute their mean to

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.4: Distributions Of Data

Problem 19PFA

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Question

A. What is the best point estimate, based on the sample, to use for the population mean? (Answer is in seconds)

B. For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 99% confidence interval for the population mean. (See picture)

### Sampling Scenarios: Choosing the Appropriate Statistical Test

Below is a table summarizing various sampling scenarios and the appropriate tests to use (Z or t). The scenarios take into consideration the sample size, distribution, and whether the standard deviation is known.

| **Sampling scenario** | **Z** | **t** | **Could use either $ Z $ or $ t $** | **Unclear** |
|:----------------------|:-----:|:-----:|:---------------------------------------:|:-----------:|
| The sample has size 10, and it is from a normally distributed population with an unknown standard deviation. | ⃝ | ⃝ | ⃝ | ⃝ |
| The sample has size 90, and it is from a non-normally distributed population with a known standard deviation of 0.45. | ⃝ | ⃝ | ⃝ | ⃝ |
| The sample has size 85, and it is from a non-normally distributed population. | ⃝ | ⃝ | ⃝ | ⃝ |

### Explanation

- **First Scenario**:
- **Sampling context**: Small sample size (n=10) from a normally distributed population with an unknown standard deviation.
- **Test choice**: Typically, for small sample sizes and unknown standard deviations from normally distributed populations, the t-test is more appropriate.

- **Second Scenario**:
- **Sampling context**: Large sample size (n=90) from a non-normally distributed population with a known standard deviation (0.45).
- **Test choice**: For large sample sizes, the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal. Thus, either Z or t tests could potentially be used, but the Z-test might be preferred due to the known standard deviation.

- **Third Scenario**:
- **Sampling context**: Large sample size (n=85) from a non-normally distributed population without specification of the standard deviation.
- **Test choice**: Similar to the second scenario, a large sample size means the Central Limit Theorem can be applied, and thus either test can often be used.

The circles (⃝) are placeholders indicating the options evaluated in each scenario. For educational purposes, each scenario should be further analyzed to determine the most precise statistical test based on given conditions

Definition Definition Method in statistics by which an observation’s uncertainty can be quantified. The main use of interval estimating is for describing a range that is made by transforming a point estimate by determining the range of values, or interval within which the population parameter is likely to fall. This range helps in measuring its precision.

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