π€ GPT

Need help solving a hypothesis testing problem? This prompt allows you to input all the essential details, such as the type of test (t-test, z-test, chi-square, etc.), your null and alternative hypotheses, sample data, population parameters, and significance level. Whether you're looking for a step-by-step solution or specific outputs like p-values, confidence intervals, or decision rules, this tailored approach provides a clear, accurate resolution. Optionally, you can even request the solution

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$2.99

Model

Chat - **GPT-4o **(gpt-4o)

Token size

322 ($0.01610 / call)

Example input

[Specify the test: one-tailed, two-tailed, and the type of test, e.g., t-test, z-test, chi-square test, etc.]: Two-tailed t-test
[State the null hypothesis, e.g., 'The mean is equal to X' or 'There is no relationship between variables']: The mean test score of students who received extra tutoring is equal to the mean score of those who did not
[State the alternative hypothesis, e.g., 'The mean is not equal to X' or 'There is a relationship between variables']: The mean test score of students who received extra tutoring is different from those who did not
[Provide the sample data, such as sample mean, sample size, standard deviation, or any other relevant statistics]: Group 1 (received tutoring): Sample mean = 78, Sample size = 30, Standard deviation = 8. Group 2 (no tutoring): Sample mean = 74, Sample size = 30, Standard deviation = 10
[Include population mean, standard deviation, or proportion if available or required]: Population parameters are unknown
[Specify the significance level, e.g., 0.05 or 0.01]: 0.05
[Specify what you'd like to conclude, such as p-value, confidence interval, critical value, rejection region, or decision on rejecting/accepting Hβ]: p-value, critical value, decision on rejecting/accepting Hβ
[Do you want the output in code? If so, specify which programming language: No, Python, R...]: Python

Example output

Sure! Letβs go step by step to solve this two-tailed t-test problem for comparing the means of two independent groups.
Step 1: Identify the given information
We are given the following information:
Group 1 (received tutoring):
Sample mean (
π
Λ
1
X
Λ
1
β
) = 78
Sample size (
π
1
n
1
β
) = 30
Sample standard deviation (
π
1
s
1
β
) = 8
Group 2 (no tutoring):
Sample mean (
π
Λ
2
X
Λ
2
β
) = 74
Sample size (
π
2
n
2
β
) = 30
Sample standard deviation (
π
2
s
2
β
) = 10
Significance level (
πΌ
Ξ±) = 0.05
Two-tailed t-test: Null hypothesis (
π»
0
H
0
β
):
π
1
=
π
2
ΞΌ
1
β
=ΞΌ
2
β
Step 2: Formulate the hypotheses
Null hypothesis (
π»
0
H
0
β
): The mean test score of students who received extra tutoring is equal to the mean score of those who did not. (
π
1
=
π
2
ΞΌ
1
β
=ΞΌ
2
β
).
Alternative hypothesis (
π»
1
H
1
β
): The mean test score of students who received extra tutoring is different from those who did not. (
π
1
β
π
2
ΞΌ
1
β
ξ
=ΞΌ
2
β
).
This is a two-tailed test.
Step 3: Calculate the test statistic
We will use the formula for the test statistic in a two-sample t-test, which is:
π‘
=
π
Λ
1
β
π
Λ
2
π
1
2
π
1
+
π
2
2
π
2
t=
n
1
β
s
1
2
β
β
+
n
2
β
s
2
2
β
β
β
X
Λ
1
β
β
X
Λ
2
β
β
Where:
π
Λ
1
X
Λ
1
β
and
π
Λ
2
X
Λ
2
β
are the sample means,
π
1
s
1
β
and
π
2
s
2
β
are the sample standard deviations,
π
1
n
1
β
and
π
2
n
2
β
are the sample sizes.
Step 4: Degrees of freedom
The degrees of freedom for the t-test can be approximated using the following formula (Welch's t-test formula for unequal variances):
π
π
=
(
π
1
2
π
1
+
π
2
2
π
2
)
2
(
π
1
2
π
1
)
2
π
1
β
1
+
(
π
2
2
π
2
)
2
π
2
β
1
df=
n
1
β
β1
(
n
1
β
s
1
2
β
β
)
2
β
+
n
2
β
β1
(
n
2
β
s
2
2
β
β
)
2
β
(
n
1
β
s
1
2
β
β
+
n
2
β
s
2
2
β
β
)
2
β
Step 5: Decision Rule
For a two-tailed test with a significance level (
πΌ
=
0.05
Ξ±=0.05), we need to find the critical t-value. The null hypothesis will be rejected if the absolute value of the test statistic is greater than the critical value.
Step 6: Calculate the p-value
The p-value will be calculated based on the test statistic and degrees of freedom. If the p-value is less than the significance level (
πΌ
=
0.05
Ξ±=0.05), we reject the null hypothesis.
Step 7: Conclusion
Compare the p-value with
πΌ
Ξ± and make a decision whether to reject or fail to reject the null hypothesis.
Now, let's perform these calculations using Python.
python
Copiar cΓ³digo
import math
from scipy import stats
# Given data
mean1 = 78 # Group 1 (received tutoring)
std1 = 8
n1 = 30
mean2 = 74 # Group 2 (no tutoring)
std2 = 10
n2 = 30
alpha = 0.05
# Step 3: Calculate the test statistic
numerator = mean1 - mean2
denominator = math.sqrt((std1**2 / n1) + (std2**2 / n2))
t_statistic = numerator / denominator
# Step 4: Calculate the degrees of freedom
df_numerator = ((std1**2 / n1) + (std2**2 / n2))**2
df_denominator = ((std1**2 / n1)**2 / (n1 - 1)) + ((std2**2 / n2)**2 / (n2 - 1))
df = df_numerator / df_denominator
# Step 5: Calculate the critical t-value (two-tailed)
critical_value = stats.t.ppf(1 - alpha/2, df)
# Step 6: Calculate the p-value
p_value = 2 * (1 - stats.t.cdf(abs(t_statistic), df))
# Step 7: Conclusion
reject_null = p_value < alpha
# Output results
t_statistic, df, critical_value, p_value, reject_null
Step-by-Step Explanation:
Test Statistic: The code calculates the test statistic using the formula for two independent samples.
Degrees of Freedom (df): The Welch-Satterthwaite formula is used to calculate the approximate degrees of freedom.
Critical Value: The critical value is determined from the t-distribution for a two-tailed test at
πΌ
=
0.05
Ξ±=0.05.
p-value: The p-value is calculated based on the test statistic and degrees of freedom.
Decision: We check if the p-value is less than
πΌ
Ξ± to determine whether to reject the null hypothesis.
Let me know if you'd like me to run the code or explain any further details!

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