Explain why the t-distribution has less spread as the number of degrees of freedom increases.

Question: Explain why the t-distribution has less spread as the number of degrees of freedom increases.

The t-distribution, often used when the population variance is unknown, tends to have a wider spread compared to the normal distribution, especially with fewer degrees of freedom. This wider spread reflects the added uncertainty inherent in estimating the population variance from a small sample size. As the number of degrees of freedom increases, which corresponds to a larger sample size, the sample variance becomes a more accurate estimate of the population variance. Consequently, the t-distribution becomes more peaked and less spread out, approaching the shape of the normal distribution. This transition occurs because, with more data, there's less variability and uncertainty, leading to a tighter confidence interval around the mean. In essence, as the degrees of freedom grow, the t-distribution converges on the normal distribution, reflecting increased precision in statistical estimates.

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अज्ञेय का वैवाहिक जीवन उतार-चढ़ाव भरा था, वर्णन कीजिए? Aage ka vaivahik jivan utar chadhav bhara tha varnan kijiye

सवाल: अज्ञेय का वैवाहिक जीवन उतार-चढ़ाव भरा था, वर्णन कीजिए? अज्ञेय का वैवाहिक जीवन उतार-चढ़ाव भरा था। उन्होंने दो विवाह किए, लेकिन दोनों विवाह सफल नहीं हो पाए। उनका पहला विवाह 1937 में राजकुमारी रत्नलता से हुआ था। यह विवाह अज्ञेय के परिवार की इच्छा के विरुद्ध था। अज्ञेय के पिता हीरानंद वात्स्यायन संस्कृत के विद्वान थे और वे चाहते थे कि अज्ञेय एक योग्य विदुषी से विवाह करें। रत्नलता एक साधारण परिवार से थीं और वे संस्कृत नहीं जानती थीं। इस कारण से, अज्ञेय के परिवार वालों ने इस विवाह का विरोध किया। अज्ञेय और रत्नलता का विवाह कुछ वर्षों तक चलता रहा, लेकिन अंततः उनका तलाक हो गया। इस तलाक के कई कारण थे, जिनमें अज्ञेय की व्यस्त जीवनशैली, रत्नलता की शिक्षा और सामाजिक पृष्ठभूमि में अंतर, और अज्ञेय के परिवार का विरोध शामिल था। अज्ञेय का दूसरा विवाह 1956 में कपिला मलिक से हुआ था। कपिला एक मशहूर संगीतकार थीं और वे अज्ञेय की साहित्यिक प्रतिभा की प्रशंसक थीं। यह विवाह कुछ वर्षों तक चलता रहा, लेकिन अंततः यह भी विफल हो गया। इस विफलता के कारणों में अज्ञेय के व्यस्त जीवनशैली, कपिला की स्वतंत्रताप्रियता,...

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