Annual high temperatures in a certain location have been tracked for a sample of 11 years. Let X represent the year and Y the high temperature (in C°). 4 28.6 5 30.2 6 35.6 7 37.3 38.4 44.6 10 44.8 48.1 11 12 51.5 13 55.3 14 57.8 Calculate the correlation coefficient. (Round to three decimal piaces.) Let's test the significance of the correlation coefficient at a = 0.05 significance level. Hạ: There is not a significant relationship between the year and high temperature, i.e., p = 0. Hạ: There is a significant relationship between the year and high temperature, i.e., p £ 0.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
![**Annual High Temperatures Analysis**
Annual high temperatures in a certain location have been tracked for a sample of 11 years. Let \( X \) represent the year and \( Y \) the high temperature (in °C).
| \( x \) | \( y \) |
|--------|--------|
| 4 | 28.6 |
| 5 | 30.2 |
| 6 | 35.6 |
| 7 | 37.3 |
| 8 | 38.4 |
| 9 | 44.6 |
| 10 | 44.8 |
| 11 | 48.1 |
| 12 | 51.5 |
| 13 | 55.3 |
| 14 | 57.8 |
**Calculate the correlation coefficient.** *(Round to three decimal places.)*
\( r = \) [ Fill in the blank ]
---
**Let's test the significance of the correlation coefficient at \( \alpha = 0.05 \) significance level.**
**\( H_0 \):** There is not a significant relationship between the year and high temperature, i.e., \( \rho = 0 \).
**\( H_1 \):** There is a significant relationship between the year and high temperature, i.e., \( \rho \neq 0 \).
a. **Calculate the test statistic.** *Round to three decimal places.*
\( t = \) [ Fill in the blank ]
b. **Calculate the p-value.** *Round to four decimal places.*
p-value = [ Fill in the blank ]
c. **Make a decision.**
- [ ] Do not reject the null hypothesis.
- [ ] Reject the null hypothesis.
d. **Is there enough evidence to support that there is a significant relationship between the year and high temperature?**
- [ ] No
- [ ] Yes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9849c3b-f0ff-47eb-829a-2c91d3fc6b02%2F35dbf0f8-f719-43f5-b935-f04c4582b306%2Fxo0d8a_processed.png&w=3840&q=75)
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