A auto maker is interested in information about how long transmissions last. A sample of transmissions are run constantly and the number of miles before the transmission fails is recorded. The auto maker claims that the transmissions can run constantly for over 150,000 miles before failure. The results of the sample are given below. Miles (1000s of miles) Mean 150.7 Variance 4.551 Observations 42 Hypothesized Mean 150 df 41 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail 2.17 p= Ex: 1.234 0.018 1.683 0.036 -2 -1 2 3 2.02 t = Ex: 1.234 Confidence Level(95.0%) 0.665
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![Title: Statistical Analysis of Transmission Longevity
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### Overview
An automaker is interested in understanding how long their transmissions last. For this study, a sample of transmissions was tested by running them continuously until they failed. The number of miles before the transmissions failed was recorded. The auto maker claims that the transmissions can run constantly for over 150,000 miles before failure. The results of the sample are provided below:
### Sample Data
| **Miles (1000s of miles)** | |
|----------------------------|---------|
| **Mean** | 150.7 |
| **Variance** | 4.551 |
| **Observations** | 42 |
| **Hypothesized Mean** | 150 |
| **df** (Degrees of Freedom) | 41 |
| **t Stat** | 2.17 |
| **p(T<=t) one-tail** | 0.018 |
| **t Critical one-tail** | 1.683 |
| **p(T<=t) two-tail** | 0.036 |
| **t Critical two-tail** | 2.02 |
| **Confidence Level (95.0%)**| 0.665 |
### Histogram Explanation
The image includes a graphical representation of transmission longevity data:
- A bell curve (normal distribution) is shown, centered around 0.
- The critical regions of the graph are shaded to show the p-value (probability value) regions corresponding to different t-values.
On the horizontal axis:
- The t-values range from -3 to 3.
- There is a notable point where \( t = 2.17 \), corresponding to the sample data.
### Use of T-distribution
#### What condition is met for the use of a t-distribution?
1. **The population is stated to be not skewed.**
2. **The sample size is larger than 30.**
### Calculations
To support these findings in your analysis, you may need the following formulas:
1. **Sample Size**:
\[ n = 42 \]
2. **Sample Mean**:
\[ \bar{x} = 150.7 \]
3. **Sample Variance**:
\[ s^2 = 4.551 \]
4. **Degrees of Freedom**:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52bd3cfe-3c9b-40ce-8732-3b0b81ba9280%2F18ecbb09-6492-4192-8925-3ee30569e094%2Faevi0st_processed.jpeg&w=3840&q=75)
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