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# Single-Parameter Bootstrap Test of Significance

• Bootstrap methods assume that our sample is an SRS and will give trustworthy conclusions only if this condition is met.

• These methods make no assumptions about the distribution your data comes from.

• For inference for a single population mean, single-parameter bootstrap tests are an alternative to the one-sample $t$-test when the guidelines for its use are not met (such as when the data is strongly skewed or has outliers with low sample size).

• For inference for a single population proportion, single-parameter bootstrap tests are an alternative to the one-sample $z$-test when the guidelines for its use are not met, that is when the number of successes and failures are not large enough.
 Data Frequencies Sample data goes here (enter numbers in columns): Number of Successes Sample Size Test for a: Population Mean Population Proportion Population Median Population Standard Deviation $H_0: \mu =$ $=\mu_0$ $H_a:\mu$ $\neq$ $\lt$ $\gt$ $\mu_0$ Level of Significance: $\alpha=$ 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 Number of Bootstrap Samples: 1000 5000 10,000 50,000 100,000 500,000 1,000,000 Input data as Frequency Table:

## Population Mean

 Sample Size: $n=$ Sample Mean: $\overline{x}=$ Lower Critical Value: $\mu_{L}^*=$ Upper Critical Value: $\mu_{U}^*=$ $p\mbox{-value}$:

 Frequency Sample Data Bootstrap Means $\mu^*$

## Population Proportion

 Sample Size: $n=$ Sample Proportion: $\hat{p}=$ Lower Critical Value: $p_{L}^*=$ Upper Critical Value: $p_{U}^*=$ $p\mbox{-value}$:

 Frequency Failures and Successes Bootstrap Proportions $p^*$

## Population Median

 Sample Size: $n=$ Sample Median: $M=$ Lower Critical Value: $M_{L}^*=$ Upper Critical Value: $M_{U}^*=$ $p\mbox{-value}$:

 Frequency Sample Data Bootstrap Medians $M^*$

## Population Standard Deviation

 Sample Size: $n=$ Sample Standard Deviation: $s=$ Lower Critical Value: $\sigma_{L}^*=$ Upper Critical Value: $\sigma_{U}^*=$ $p\mbox{-value}$:

 Frequency Sample Data Bootstrap Standard Deviations $\sigma^*$