One sample and two sample test t for Means

One sample & two sample test t for Means (माध्य के लिए एक-नमूना t-परीक्षण और दो-नमूना t-परीक्षण):- The t-test is a statistical test used to determine if there is a significant difference between the means of two groups or if a single group's mean significantly differs from a known value. It’s especially useful when the sample size is small and the population standard deviation is unknown. There are two main types of t-tests: the one-sample t-test and the two-sample t-test.

(t-परीक्षण एक सांख्यिकीय परीक्षण है जिसका उपयोग यह निर्धारित करने के लिए किया जाता है कि दो समूहों के औसत में कोई महत्वपूर्ण अंतर है या नहीं, या एकल समूह का औसत एक ज्ञात मान से महत्वपूर्ण रूप से भिन्न है या नहीं। यह विशेष रूप से तब उपयोगी होता है जब नमूने का आकार छोटा हो और जनसंख्या का मानक विचलन अज्ञात हो। t-परीक्षण के दो मुख्य प्रकार हैं: एक-नमूना t-परीक्षण और दो-नमूना t-परीक्षण।)

One-Sample t-Test (एक-नमूना t-परीक्षण):-

Purpose (उद्देश्य):- The one-sample t-test is used to compare the mean of a single sample to a known value or a theoretical expectation (e.g., population mean).

[एक-नमूना t-परीक्षण का उपयोग किसी एकल नमूने के औसत की तुलना एक ज्ञात मान या सैद्धांतिक अपेक्षा (जैसे जनसंख्या माध्य) से करने के लिए किया जाता है।]

When to use (कब उपयोग करें):-

> When you have one group of data.

(जब आपके पास डेटा का एक समूह हो।)

> When you want to compare the sample mean to a known or hypothesized population mean.

(जब आप नमूने के माध्य की तुलना एक ज्ञात या परिकल्पित जनसंख्या माध्य से करना चाहते हैं।)

Hypotheses (परिकल्पना):-

i. Null hypothesis (शून्य परिकल्पना) (H0):- The sample mean is equal to the population mean (μ = μ0​).

[नमूने का माध्य जनसंख्या के माध्य के बराबर है (μ=μ0)।]

ii. Alternative hypothesis (वैकल्पिक परिकल्पना) (Ha):-

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