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You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.

It's worth reiterating @mkt's point: both point & interval estimates rely on the sampling scheme's being adequately modelled for their validity. If e.g. 424 names were drawn from a hat, & all those patients tested, SRS would be an entirely appropriate model; if e.g. 424 consented to the antibody test out of 618 asked, you'd have to make a case based on clinical data for the 194 patients that refused and their stated reason for refusal that they were neither more not less likely to test positive for COVID-19 antibodies than patients who consented.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.

You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.

It's worth reiterating @mkt's point: both point & interval estimates rely on the sampling scheme's being adequately modelled for their validity. If e.g. 424 names were drawn from a hat, & all those patients tested, SRS would be an entirely appropriate model; if e.g. 424 consented to the antibody test out of 618 asked, you'd have to make a case based on clinical data for the 194 patients that refused, & on their stated reason for refusal, that they were neither more nor less likely to test positive for COVID-19 antibodies than patients who consented.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.

You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.

You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.

It's worth reiterating @mkt's point: both point & interval estimates rely on the sampling scheme's being adequately modelled for their validity. If e.g. 424 names were drawn from a hat, & all those patients tested, SRS would be an entirely appropriate model; if e.g. 424 consented to the antibody test out of 618 asked, you'd have to make a case based on clinical data for the 194 patients that refused and their stated reason for refusal that they were neither more not less likely to test positive for COVID-19 antibodies than patients who consented.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.

You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.

You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the confidence interval may well be wider than you wanted—but not that it's untrustworthy.^†

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.

† You may have more reason to use exact methods rather than an asymptotic approximation, however.