# t-test and z-test calculator

> t-test calculator for one-sample, two-sample (Welch or pooled) and paired tests, plus z-tests for a mean or proportion: p-value, critical value, decision.

ઇન્ટરએક્ટિવ આવૃત્તિ: https://www.calcopenly.com/gu/statistics/t-test-calculator
વિષય: આંકડાશાસ્ત્ર અને સંભાવનાના કેલ્ક્યુલેટર

A t-test divides the gap between a sample mean and a hypothesised value, or between two sample means, by its standard error. The ratio follows Student's t distribution with n − 1 degrees of freedom for one sample or for paired differences, n₁ + n₂ − 2 for the pooled two-sample test, and the Welch–Satterthwaite value when the variances may differ. With a known population σ the same ratio is a z statistic; the proportion test compares a sample proportion with p₀ the same way.

Quality checks, A/B tests and before-and-after studies are typical uses. The default data are 12 fill weights tested against a target of 500: the mean is 498.64, t = −2.395 on 11 df and the two-sided p = 0.0355, so the mean differs from 500 at α = 0.05.

The t-tests assume independent observations from roughly normal populations. Larger samples tolerate more skew, because the distribution of the sample mean approaches normal as n grows.

## ઇનપુટ

- **Test** (વિકલ્પો: One-sample t-test, One-sample z-test (σ known), Two-sample t-test, Welch (unequal variances), Two-sample t-test, pooled variance, Paired t-test, One-proportion z-test)
- **Enter** (વિકલ્પો: Raw data, Summary statistics)
- **Sample 1**: For a paired test, the first measurement of each pair.
- **Sample 2**: For a paired test, the second measurement, in the same order.
- **Sample 1 mean**: For a paired test, the mean of the differences.
- **Standard deviation s₁**: For a paired test, the SD of the differences.
- **Sample size n₁**
- **Sample 2 mean**
- **Standard deviation s₂**
- **Sample size n₂**
- **Known population σ**
- **Hypothesised mean μ₀**
- **Hypothesised difference Δ₀**: Mean of sample 1 minus sample 2 under H₀ (usually 0).
- **Successes x**
- **Sample size n**
- **Hypothesised proportion p₀**
- **Alternative hypothesis** (વિકલ્પો: ≠ (two-sided), < (left-tailed), > (right-tailed))
- **Significance level α**

## પરિણામો

- p-value — મુખ્ય પરિણામ
- Decision
- Test statistic
- Degrees of freedom
- Critical value
- Estimate
- Standard error

## સૂત્ર

$$
t = \frac{\bar x - \mu_0}{s/\sqrt n},\qquad t_{\text{Welch}} = \frac{\bar x_1 - \bar x_2 - \Delta_0}{\sqrt{s_1^2/n_1 + s_2^2/n_2}},\qquad z = \frac{\hat p - p_0}{\sqrt{p_0(1-p_0)/n}}
$$

## ઉકેલેલાં ઉદાહરણો

### One-sample t on fill weights (defaults)

- Test: One-sample t-test
- Enter: Raw data
- Sample 1: 498.2, 501.3, 497.6, 499.1, 495.8, 500.4, 496.9, 498.7, 497.3, 499.8, 502.1, …
- Hypothesised mean μ₀: 500
- Alternative hypothesis: ≠ (two-sided)
- Significance level α: 0.05
- **Test statistic: -2.395294**
- **Degrees of freedom: 11**
- **p-value: 0.035527**
- **Decision: Reject H₀**
- **Critical value: 2.201**
- ચકાસણીનો સ્ત્રોત: Python statistics.mean/stdev for t; p-value from the A&S 26.7.3 closed form for odd ν (pyref.t_cdf_int); t₀.₉₇₅,₁₁ = 2.201 (t table)

### Welch t from summary statistics

- Test: Two-sample t-test, Welch (unequal variances)
- Enter: Summary statistics
- Sample 1 mean: 20.1
- Standard deviation s₁: 3.2
- Sample size n₁: 15
- Sample 2 mean: 17.4
- Standard deviation s₂: 4.8
- Sample size n₂: 12
- Hypothesised difference Δ₀: 0
- Alternative hypothesis: ≠ (two-sided)
- Significance level α: 0.05
- **Test statistic: 1.673611**
- **Degrees of freedom: 18.3865**
- **p-value: 0.111135**
- ચકાસણીનો સ્ત્રોત: Welch–Satterthwaite df in Python fractions; p-value by Gauss–Legendre quadrature of the t density (pyref.t_sf_numeric)

### Pooled t on the same summaries

- Test: Two-sample t-test, pooled variance
- Enter: Summary statistics
- Sample 1 mean: 20.1
- Standard deviation s₁: 3.2
- Sample size n₁: 15
- Sample 2 mean: 17.4
- Standard deviation s₂: 4.8
- Sample size n₂: 12
- Hypothesised difference Δ₀: 0
- Alternative hypothesis: ≠ (two-sided)
- Significance level α: 0.05
- **Test statistic: 1.749856**
- **Degrees of freedom: 25**
- **p-value: 0.09241**
- ચકાસણીનો સ્ત્રોત: Pooled variance in Python fractions; p-value from the A&S 26.7.3 closed form with ν = 25

### Paired t, right-tailed

- Test: Paired t-test
- Enter: Raw data
- Sample 1: 142 138 150 145 160 155 139 148
- Sample 2: 136 135 146 144 150 149 138 141
- Hypothesised difference Δ₀: 0
- Alternative hypothesis: > (right-tailed)
- Significance level α: 0.05
- **Test statistic: 4.32649**
- **Degrees of freedom: 7**
- **p-value: 0.001726**
- **Estimate: 4.75**
- ચકાસણીનો સ્ત્રોત: Differences' mean and stdev in Python statistics; p-value from the A&S closed form with ν = 7

### One-proportion z, 540 of 1000 vs 0.5

- Test: One-proportion z-test
- Successes x: 540
- Sample size n: 1000
- Hypothesised proportion p₀: 0.5
- Alternative hypothesis: ≠ (two-sided)
- Significance level α: 0.05
- **Test statistic: 2.529822**
- **p-value: 0.011412**
- **Decision: Reject H₀**
- ચકાસણીનો સ્ત્રોત: z = 0.04/√(0.25/1000); p = erfc(z/√2) in Python

### One-sample z, right-tailed

- Test: One-sample z-test (σ known)
- Enter: Summary statistics
- Sample 1 mean: 103
- Sample size n₁: 36
- Known population σ: 15
- Hypothesised mean μ₀: 100
- Alternative hypothesis: > (right-tailed)
- Significance level α: 0.05
- **Test statistic: 1.2**
- **p-value: 0.11507**
- **Critical value: 1.6449**
- ચકાસણીનો સ્ત્રોત: z table: 1 − Φ(1.20) = 0.1151; Python 0.5·erfc(1.2/√2) = 0.1150697; z₀.₉₅ = 1.644854

## પ્રશ્નો

### What does a p-value tell you?

It is the probability of a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. For the default data p = 0.0355: if the true mean were 500, samples at least this far from 500 would turn up about 3.6% of the time. It is not the probability that H₀ is true and it does not measure effect size, as the American Statistical Association's 2016 statement on p-values stresses.

### Should I use Welch's t-test or the pooled t-test?

Use Welch's test unless you have good reason to believe the variances are equal. It drops the equal-variance assumption and loses little power when the variances do match, while the pooled test's false-positive rate drifts from α when variances and group sizes both differ. On the worked example (s = 3.2 and 4.8, n = 15 and 12) Welch gives p = 0.111 on 18.4 df and the pooled test p = 0.092 on 25 df.

### When should I use a paired t-test?

When each value in one sample is matched to one in the other: the same patients before and after treatment, or two instruments measuring the same parts. The test is a one-sample t-test on the differences, which removes the variation between subjects. In the paired worked example eight pairs differ by 4.75 on average, giving t = 4.33 on 7 df and a one-sided p of 0.0017.

### What is the difference between a t-test and a z-test?

A z-test uses a known population standard deviation σ and the standard normal distribution; a t-test estimates σ from the sample and uses Student's t, whose heavier tails allow for that extra uncertainty. The two-sided 5% critical value is 1.960 for z, 2.201 for t with 11 df and 2.042 with 30 df. σ is rarely known in practice, so the t-test is the usual choice for means.

### Should I use a one-tailed or two-tailed test?

Use a two-tailed test unless the direction was fixed before seeing the data and an effect in the other direction would be treated the same as no effect. A one-tailed test puts all of α in one tail, so its p-value is half the two-tailed one when the effect goes the predicted way: t = −2.395 on 11 df gives 0.0355 two-tailed and 0.0178 left-tailed. Picking the tail after looking doubles the real false-positive rate.

### “t-test and z-test calculator” કેટલું ચોક્કસ છે?

ચોકસાઈ તમારા ઇનપુટ અને પદ્ધતિની ધારણાઓ પર આધારિત છે. દશાંશ ગણતરી 50 સાર્થક અંકો વાપરે છે, પરંતુ અંદાજ, સંખ્યાત્મક પદ્ધતિઓ અને મૂળ ડેટા ઓછા ચોક્કસ હોઈ શકે છે; દર્શાવેલા મૂલ્યોને રાઉન્ડ કરવાથી આ મર્યાદાઓ દૂર થતી નથી. સ્વતંત્ર સ્ત્રોતોના ઉકેલેલા ઉદાહરણો સાથે ચકાસણી: 7. ઉદાહરણ તરીકે, “One-sample t on fill weights (defaults)”ને Python statistics.mean/stdev for t; p-value from the A&S 26.7.3 closed form for odd ν (pyref.t_cdf_int); t₀.₉₇₅,₁₁ = 2.201 (t table) સાથે ચકાસવામાં આવે છે.

### આ પદ્ધતિનો સ્ત્રોત શું છે?

NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-sample t-test for equal means; NIST/SEMATECH e-Handbook, §7.2.2 Are the data consistent with the assumed process mean?; Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika 34, 28–35.

## સ્રોતો

- [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-sample t-test for equal means](https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm)
- [NIST/SEMATECH e-Handbook, §7.2.2 Are the data consistent with the assumed process mean?](https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm)
- Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika 34, 28–35